By ilikeafrica in arXiv — 19 Nov 2025

ON THE GENERAL STRUCTURE OF HAMILTONIAN REDUCTIONS

이 논문은 Wess-Zumino-Novikov-Witten(WZNW) 이론의 하밀턴 해소에 대한 일반적인 구조를 분석한다. 연구진은 Lie 대수 조건을 제시하여 정확한 적분성, 유사변형 불변성 및 W-동작이 존재하는 감축된 이론에서 정교하고 통합적인 적분적 그리고 BRST 공식 체계를 구축한다. 연구 결과는 수식어법과 시공간의 하밀턴 해소에 대한 일반적인 구조와 관련이 있다.

WZNW 이론은 Lie 대수에 기반한 가장 경제적인 표현으로, 그의 공역은 주로 왼쪽 × 오른쪽 Kac-Moody(KM) 공역의 직접 곱이다. WZNW → Toda 하밀턴 해소는 KM 유효성을 가진 특정한 첫 번째 등급 제약을 적용하여 KM 유동체를 W-대수에 기반한 공역으로 감축한다.

이 연구는 W-대수의 일반화와 Lie 대수의 완전성 조건에 대한 깊은 통찰력을 제공하며, 연구 결과가 여러 예제를 거쳐 테스트되었다.

영어 요약 시작:

ON THE GENERAL STRUCTURE OF HAMILTONIAN REDUCTIONS

arXiv:hep-th/9112068v1 22 Dec 1991DIAS-STP-91-29UdeM-LPN-TH-71/91ON THE GENERAL STRUCTURE OF HAMILTONIAN REDUCTIONSOF THE WZNW THEORYL. Feh´er*, L. O’Raifeartaigh, P. Ruelle and I. TsutsuiDublin Institute for Advanced Studies10 Burlington Road, Dublin 4, IrelandA.

WipfInstitut f¨ur Theoretische PhysikEidgen¨ossische Technische HochschuleH¨onggerberg, Z¨urich CH-8093, SwitzerlandAbstractThe structure of Hamiltonian reductions of the Wess-Zumino-Novikov-Witten(WZNW) theory by ﬁrst class Kac-Moody constraints is analyzed in detail. Lie alge-braic conditions are given for ensuring the presence of exact integrability, conformalinvariance and W-symmetry in the reduced theories.

A Lagrangean, gauged WZNWimplementation of the reduction is established in the general case and thereby the pathintegral as well as the BRST formalism are set up for studying the quantum version ofthe reduction. The general results are applied to a number of examples.

In particular, aW-algebra is associated to each embedding of sl(2) into the simple Lie algebras by usingpurely ﬁrst class constraints. The importance of these sl(2) systems is demonstrated byshowing that they underlie the W ln-algebras as well.

New generalized Toda theories arefound whose chiral algebras are the W-algebras belonging to the half-integral sl(2) em-beddings, and the W-symmetry of the eﬀective action of those generalized Toda theoriesassociated with the integral gradings is exhibited explicitly. * Present address:Laboratoire de Physique Nucl´eaire, Universit´e de Montr´eal,Montr´eal, Canada H3C 3J7.

e-mail: feher@lps.umontreal.ca.bitnet1

Contents1. Introduction .

. .

32. General structure of KM and WZNW reductions .

. .

. 112.1.

First class and conformally invariant KM constraints . .

. .

112.2. Lagrangean realization of the Hamiltonian reduction .

. .

. 152.3.

Eﬀective ﬁeld theories from left-right dual reductions . .

. .

. .183.

Polynomiality in KM reductions and the WGS -algebras . .

. .

. 263.1.

A suﬃcient condition for polynomiality . .

. .

. 263.2.

The polynomiality of the Dirac bracket . .

. .

. 313.3.

First class constraints for the WGS -algebras . .

. .

343.4. The WGS interpretation of the W ln-algebras .

. .

. 414.

Generalized Toda theories . .

. .

.464.1. Generalized Toda theories associated with integral gradings .

. .

464.2. Generalized Toda theories for half-integral sl(2) embeddings .

. .

. .484.3.

Two examples of generalized Toda theories . .

. .

. .535.

Quantum framework for WZNW reductions . .

. .

. 575.1.

Path-integral for constrained WZNW theory . .

. .

575.2. Eﬀective theory in the physical gauge .

. .

605.3. The W-symmetry of the generalized Toda action IHeﬀ(b) .

. .

625.4. BRST formalism for WZNW reductions .

. .

. .645.5.

The Virasoro centre in two examples . .

. .

. .676.

Discussion . .

. .

70AppendicesA: A solvable but not nilpotent gauge algebra . .

. .

. 73B: H-compatible sl(2) and the non-degeneracy condition .

. .

. 77C: H-compatible sl(2) embeddings and halvings .

. .

80References . .

. .

. 872

1. IntroductionDue to their intimate relationship with Lie algebras, the various one- and two-dimensional Toda systems are among the most important models of the theory of in-tegrable non-linear equations [1-19].

In particular, the standard conformal Toda ﬁeldtheories, which are given by the LagrangeanLToda(ϕ) = κ2lXi,j=112|αi|2 Kij∂µϕi∂µϕj −lXi=1m2i expn12lXj=1Kijϕjo,(1.1)where κ is a coupling constant, Kij is the Cartan matrix and the αi are the simpleroots of a simple Lie algebra of rank l, have been the subject of many studies [1,3,4,8-13,19]. It has been ﬁrst shown by Leznov and Saveliev [1,3] that the Euler-Lagrangeequations of (1.1) can be written as a zero curvature condition, are exactly integrable,and possess interesting non-linear symmetry algebras [3,4,10,11,13,19].

These symmetryalgebras are generated by chiral conserved currents, and are polynomial extensions ofthe chiral Virasoro algebras generated by the traceless energy-momentum tensor. Thechiral currents in question are conformal primary ﬁelds, whose conformal weights aregiven by the orders of the independent Casimirs of the corresponding simple Lie algebra.Polynomial extensions of the Virasoro algebra by chiral primary ﬁelds are generally knownas W-algebras [20], which are expected to play an important role in the classiﬁcationof conformal ﬁeld theories and are in the focus of current investigations [20-29].

Theimportance of Toda systems in two-dimensional conformal ﬁeld theory is in fact greatlyenhanced by their realizing the W-algebra symmetries.It has been discovered recently that the conformal Toda ﬁeld theories can be nat-urally viewed as Hamiltonian reductions of the Wess-Zumino-Novikov-Witten (WZNW)theory [12,13]. The main feature of the WZNW theory is its aﬃne Kac-Moody (KM)symmetry, which underlies its integrability [30,31].The WZNW theory provides themost ‘economical’ realization of the KM symmetry in the sense that its phase space isessentially a direct product of the left × right KM phase spaces.

The WZNW →TodaHamiltonian reduction is achieved by imposing certain ﬁrst class, conformally invari-ant constraints on the KM currents, which reduce the chiral KM phase spaces to phasespaces carrying the chiral W-algebras as their Poisson bracket structure [12,13]. Thusthe W-algebra is related to the phase space of the Toda theory in the same way as theKM algebra is related to the phase space of the WZNW theory.

In the above manner,the W-symmetry of the Toda theories becomes manifest by describing these theories as3

reduced WZNW theories. This way of looking at Toda theories has also numerous otheradvantages, described in detail in [13].The constrained WZNW (KM) setting of the standard Toda theories (W-algebras)allows for generalizations, some of which have already been investigated [14-18,26-29].An important recent development is the realization that it is possible to associate ageneralized W-algebra to every embedding of the Lie algebra sl(2) into the simple Liealgebras [16-18].

The standard W-algebra, occurring in Toda theory, corresponds to theso called principal sl(2). In fact, these generalized W-algebras can be obtained fromthe KM algebra by constraining the current to the highest weight gauge, which hasbeen originally introduced in [13] for describing the standard case.

Another interestingdevelopment is the W ln-algebras introduced by Bershadsky [26] and further studied in[28]. It is known that the simplest non-trivial case W 23 , which was originally proposed byPolyakov [27], falls into a special case of the W-algebras obtained by the sl(2) embeddingsmentioned above.

It has not been clear, however, as to whether the two classes of W-algebras are related in general, or to what extent one can further generalize the KMreduction to achieve new W-algebras.In the present paper, we undertake the ﬁrst systematic study of the Hamiltonianreductions of the WZNW theory, aiming at uncovering the general structure of the reduc-tion and, at the same time, try to answer the above question. Various diﬀerent questionsarising from this main problem are also addressed (see Contents), and some of themcan be examined on its own right.

As this provides our motivation and in fact most ofthe later developments originate from it, we wish to recall here the main points of theWZNW →Toda reduction before giving a more detailed outline of the content.To make contact with the Toda theories, we consider the WZNW theory*SWZ(g) = κ2Zd2x ηµν Tr (g−1∂µg)(g−1∂νg) −κ3ZB3Tr (g−1dg)3 ,(1.2)for a simple, maximally non-compact, connected real Lie group G. In other words, weassume that the simple Lie algebra, G, corresponding to G allows for a Cartan decom-position over the ﬁeld of real numbers. The ﬁeld equation of the WZNW theory can bewritten in the equivalent forms∂−J = 0or∂+ ˜J = 0 ,(1.3)*The KM level k is −4πκ.

The space-time conventions are: η00 = −η11 = 1 andx± = 12(x0 ± x1). The WZNW ﬁeld g is periodic in x1 with period 2πr.4

whereJ = κ∂+g · g−1 ,and˜J = −κg−1∂−g . (1.4)These equations express the conservation of the left- and right KM currents, J and ˜J,respectively.

The general solution of the WZNW ﬁeld equation is given by the simpleformulag(x+, x−) = gL(x+) · gR(x−) ,(1.5)where gL and gR are arbitrary G-valued functions, i.e., constrained only by the boundarycondition imposed on g.Let now M−, M0 and M+ be the standard generators of the principal sl(2) subalge-bra of G [32]. By considering the eigenspaces Gm of M0 in the adjoint of G, adM0 = [M0, ],one can deﬁne a grading of G by the eigenvalues m. Under the principal sl(2) this gradingis an integral grading, in fact the spins occurring in the decomposition of the adjoint ofG are the exponents of G, which are related to the orders of the independent Casimirsby a shift by 1.

It is also worth noting that the grade 0 part ofG = G+ + G0 + G−,G± =NXm=1G±m ,(1.6)is a Cartan subalgebra, and (by using some automorphism of the Lie algebra) one canassume that the generator M0 is given by the formula M0 = 12Pα>0 Hα, where Hα is thestandard Cartan generator corresponding to the positive root α, and the generators M±are certain linear combinations of the step operators E±αi corresponding to the simpleroots αi, i = 1, . .

., rank G.The basic observation of [12,13] has been that the standard Toda theory can beobtained from the WZNW theory by imposing ﬁrst class constraints which restrict thecurrents to take the following form:J(x) = κM−+ j(x),withj(x) ∈(G0 + G+) ,(1.7a)and˜J(x) = −κM+ + ˜j(x),with˜j(x) ∈(G0 + G−) . (1.7b)(For clarity, we note that one should in principle include some dimensional constantsin M± which are dimensionless, but such constants are always put to unity in thispaper, for simplicity.) To derive the Toda theory (1.1) from the WZNW theory (1.2),one uses the generalized Gauss decomposition g = g+ · g0 · g−of the WZNW ﬁeld g,5

where g0,± are from the subgroups G0,± of G corresponding to the Lie subalgebras G0,±,respectively. In this framework the Toda ﬁelds ϕi are given by the middle-piece of theGauss decomposition, g0 = exp[ 12Pli=1 ϕiHi], which is invariant under the triangularKM gauge transformations belonging to the ﬁrst class constraints (1.7).

Note that herethe elements Hi ∈G0 are the standard Cartan generators associated to the simple roots.In fact, the Toda ﬁeld equation can be derived directly from the WZNW ﬁeld equationby inserting the Gauss decomposition of g into (1.3) and using the constraints (1.7).The eﬀective action of the reduced theory, (1.1), can also be obtained in a natural way,by using the Lagrangean, gauged WZNW implementation of the Hamiltonian reduction[13].In their pioneering work [1,3], Leznov and Saveliev proved the exact integrability ofthe conformal Toda systems by exhibiting chiral quantities by using the ﬁeld equationand the special graded structure of the Lax potential A±, in terms of which the Todaequation takes the zero curvature form[∂+ −A+ , ∂−−A−] = 0 . (1.8)In our framework the exact integrability of Toda systems is seen as an immediate con-sequence of the obvious integrability of the WZNW theory, which survives the reductionto Toda theory.

In other words, the chiral ﬁelds underlying the integrability of the Todaequation are available from the very beginning, that is, they come from the ﬁelds enteringthe left × right decomposition of the general WZNW solution (1.5). Furthermore, theToda Lax potential itself emerges naturally from the trivial, chiral Lax potential of theWZNW theory.

To see this one ﬁrst observes that the WZNW ﬁeld equation is a zerocurvature condition, since one can write for example the ﬁrst equation in (1.3) as[∂+ −J , ∂−−0] = 0 . (1.9)Using the constraints of the reduction, the Toda zero curvature condition (1.8) of [1,3]arises from (1.9) by conjugating this equation by g−1+ (x+, x−), namely by the inverse ofthe upper triangular piece of the generalized Gauss decomposition of the WZNW ﬁeld g[18].The W-symmetry of the Toda theory appears in the WZNW setting in a very directand natural way.

Namely, one can interpret the W-algebra as the KM Poisson bracketalgebra of the gauge invariant diﬀerential polynomials of the constrained currents in (1.7).Concentrating on the left sector, the gauge transformations act on the current according6

toJ(x) →ea(x+) J(x) e−a(x+) + κ(ea(x+))′ e−a(x+),(1.10)where a(x+) ∈G+ is an arbitrary chiral parameter function. * The constraints (1.7) arechosen in such a way that the following Virasoro generatorLM0(x) ≡LKM(x) −Tr (M0J′(x)),whereLKM(x) = 12κTr(J2(x)),(1.11)is gauge invariant, which ensures the conformal invariance of the reduced theory.One obtains an equivalent interpretation of the W-algebra by identifying it withthe Dirac bracket algebra of the diﬀerential polynomials of the current components incertain gauges, which are such that a basis of the gauge invariant diﬀerential polynomialsreduces to the independent current components after the gauge ﬁxing.

We call the gaugesin question Drinfeld-Sokolov (DS) gauges [13], since such gauges has been used also in[5]. They have the nice property that any constrained current J(x) can be brought to thegauge ﬁxed form by a unique gauge transformation depending on J(x) in a diﬀerentialpolynomial way.

The most important DS gauge is the highest weight gauge [13], whichis deﬁned by requiring the gauge ﬁxed current to be of the following form:Jred(x) = κM−+ jred(x) ,jred(x) ∈Ker(adM+) ,(1.12)where Ker(adM+) is the kernel of the adjoint of M+. In other words, jred(x) is restrictedto be an arbitrary linear combination of the highest weight vectors of the sl(2) subalgebrain the adjoint of G. The special property of the highest weight gauge is that in this gaugethe conformal properties become manifest.Of course, the quantity Lred(x) obtainedby restricting LM0(x) in (1.11) to the highest weight gauge generates a Virasoro algebraunder Dirac bracket.

(Note that in our case Lred(x) is proportional to the M+-componentof jred(x).) The important point is that, with the exception of the M+-component, thespin s component of jred(x) is in fact a primary ﬁeld of conformal weight (s + 1) withrespect to Lred(x) under the Dirac bracket.

Thus the highest weight gauge automaticallyyields a primary ﬁeld basis of the W-algebra, from which one sees that the spectrum ofconformal weights is ﬁxed by the sl(2) content of the adjoint of G [13].In the above we arrived at the description of the W-algebra as a Dirac bracket algebraby gauge ﬁxing the ﬁrst class system of constraints corresponding to (1.7). However, it is* Throughout the paper, the notation f ′ = 2∂1f is used for every function f, includingthe spatial δ-functions.

For a chiral function f(x+) one has then f ′ = ∂+f.7

clear now that it would have been possible to deﬁne the W-algebra as the Dirac bracketalgebra of the components of jred in (1.12) in the ﬁrst place. Once this point is realized,a natural generalization arises immediately [16-18].

Namely, one can associate a classicalW-algebra to any sl(2) subalgebra S = {M−, M0, M+} of any simple Lie algebra G, bydeﬁning it to be the Dirac bracket algebra of the components of jred in (1.12), where onesimply substitutes the generators M± of the arbitrary sl(2) subalgebra S for those of theprincipal sl(2). As we shall see in this paper, this Dirac bracket algebra is a polynomialextension of the Virasoro algebra by primary ﬁelds, whose conformal weights are relatedto the spins occurring in the decomposition of the adjoint of G under S by a shift by 1, incomplete analogy with the case of the principal sl(2).

We shall designate the generalizedW-algebra associated to the sl(2) embedding S as WGS .With the main features of the WZNW →Toda reduction and the above deﬁnitionof the WGS -algebras at our disposal, now we sketch the philosophy and the outline ofthe present paper. We start by giving the most important assumption underlying ourinvestigations, which is that we consider those reductions which can be obtained byimposing ﬁrst class KM constraints generalizing the ones in (1.7).

To be more precise,our most general constraints restrict the current to take the following form:J(x) = κM + j(x),withj(x) ∈Γ⊥,(1.13)where M is some constant element of the underlying simple Lie algebra G, and Γ⊥is thesubspace consisting of the Lie algebra elements trace orthogonal to some subspace Γ of G.We note that earlier in (1.7a) we have chosen Γ = G+ and M = M−, but we do not needany sl(2) structure here. The whole analysis is based on requiring the ﬁrst-classness ofthe system of linear KM constraints corresponding the pair (Γ, M) according to (1.13).However, this ﬁrst-classness assumption is not as restrictive as one perhaps might thinkat ﬁrst sight.

In fact, as far as we know, our ﬁrst class method is capable of covering allHamiltonian reductions of the WZNW theory considered to date. The many technicaladvantages of using purely ﬁrst class KM constraints will be apparent.The investigations in this paper are organized according to three distinct levels ofgenerality.

At the most general level we only make the ﬁrst-classness assumption anddeduce the following results.First, we give a complete Lie algebraic analysis of theconditions on the pair (Γ, M) imposed by the ﬁrst-classness of the constraints. We shallsee that Γ in (1.13) has to be a subalgebra of G on which the Cartan-Killing form vanishes,and that every such subalgebra is solvable.

The Lie subalgebra Γ will be referred to asthe ‘gauge algebra’ of the reduction. For a given Γ, the ﬁrst-classness imposes a further8

condition on the element M, and we shall describe the space of the allowed M’s. Second,we establish a gauged WZNW implementation of the reduction, generalizing the onefound previously in the standard case [13].

This gauged WZNW setting of the reductionwill be ﬁrst seen classically, but it will be also established in the quantum theory byconsidering the phase space path integral of the constrained WZNW theory. Third, thegauged WZNW framework will be used to set up the BRST formalism for the quantumHamiltonian reduction in the general case.

Fourth, by making the additional assumptionthat the left and right gauge algebras are dual to each other with respect to the Cartan-Killing form, we will be able to give a detailed local analysis of the eﬀective theoriesresulting from the reduction. This duality assumption will also be related to the parityinvariance of the eﬀective theories, which is satisﬁed in the standard Toda case where theleft and right gauge algebras are G+ and G−in (1.6), respectively.

In general, the WZNWreduction not only allows us to make contact with known theories, like the Toda theoryin (1.1), where the simplicity and the large symmetry of the ‘parent’ WZNW theory arefully exploited for analyzing them, but also leads to new theories which are ‘integrableby construction’.At the next level of generality, we study the conformally invariant reductions. Thebasic idea here is that one can guarantee the conformal invariance of the reduced theoryby exhibiting a Virasoro density such that the corresponding conformal action preservesthe constraints in (1.13).

Generalizing (1.11), we assume that this Virasoro density is ofthe formLH(x) = LKM(x) −Tr (HJ′(x)) ,(1.14)where H is some Lie algebra element, to be determined from the condition that LHweakly commutes with the ﬁrst class constraints. We shall describe the relations whichare imposed on the triple of quantities (Γ, M, H) by this requirement, and thereby obtaina Lie algebraic suﬃcient condition for conformal invariance.At the third level of generality, we deal with polynomial reductions and W-algebras.The above mentioned suﬃcient condition for conformal invariance is a guarantee for LHbeing a gauge invariant diﬀerential polynomial.

We shall provide an additional conditionon the triple of quantities (Γ, M, H) which allows one to construct out of the currentin (1.13) a complete set of gauge invariant diﬀerential polynomials by means of a poly-nomial gauge ﬁxing algorithm. The KM Poisson bracket algebra of the gauge invariantdiﬀerential polynomials yields a polynomial extension of the Virasoro algebra generatedby LH.

The most important application of our suﬃcient condition for polynomialityconcerns the WGS -algebras mentioned previously.9

Let us remember that, for an arbitrary sl(2) subalgebra S of G, the WGS -algebra canbe deﬁned as the Dirac bracket algebra of the highest weight current in (1.12) realizedby purely second class constraints. However, we shall see in this paper that these secondclass constraints can be replaced by purely ﬁrst class constraints even in the case ofarbitrary, integral or half-integral, sl(2) embeddings.Since the ﬁrst class constraintssatisfy our suﬃcient condition for polynomiality, we can realize the WGS -algebra as theKM Poisson bracket algebra of the corresponding gauge invariant diﬀerential polynomials.After having our hands on ﬁrst class KM constraints leading to the WGS -algebras, weshall immediately apply our general construction to exhibiting reduced WZNW theoriesrealizing these W-algebras as their chiral algebras for arbitrary sl(2)-embeddings.

Inthe non-trivial case of half-integral sl(2)-embeddings, these generalized Toda theoriesrepresent a new class of integrable models, which will be studied in some detail. It is alsoworth noting that realizing the WGS -algebra as a KM Poisson bracket algebra of gaugeinvariant diﬀerential polynomials should in principle allow for quantizing it through theKM representation theory, for example by using the general BRST formalism which willbe set up in this paper.

As a ﬁrst step, we shall give a concise formula for the Virasorocentre of this algebra in terms of the level of the underlying KM algebra.The existence of purely ﬁrst class KM constraints leading to the WGS -algebra mightbe perhaps surprizing to the reader, since earlier in [16] it was claimed to be inevitablynecessary to use at least some second class constraints from the very beginning, whenreducing the KM algebra to WGS in the case of a half-integral sl(2) embedding. Contraryto their claim, we will demonstrate that it is possible and in fact easy to obtain theappropriate ﬁrst class constraints which lead to WGS .

Roughly speaking, this will beachieved by discarding ‘half’ of those constraints which form the second class part in themixed system of the constraints imposed in [16]. The mixed system of constraints canbe recovered by a partial gauge ﬁxing of our purely ﬁrst class KM constraints.

Similarly,Bershadsky’s constraints [26], used to deﬁne the W ln-algebra, are also a mixed system inthe above sense, i.e., it contains both ﬁrst and second class parts. We can also replacethese constraints by purely ﬁrst class ones without changing the ﬁnal reduced phasespace.

In this procedure we shall uncover the hidden sl(2) structure of the W ln-algebras,namely, we shall identify them in general as further reductions of particular WGS -algebras.The study of WZNW reductions embraces various subjects, such as integrable mod-els, W-algebras and their ﬁeld theoretic realizations. We hope that the readers withdiﬀerent interests will ﬁnd relevant results throughout this paper, and ﬁnd an interplayof general considerations and investigations of numerous examples.10

2. General structure of KM and WZNW reductionsThe purpose of this chapter is to investigate the general structure of those reductionsof the KM phase space and corresponding reductions of the full WZNW theory whichcan be deﬁned by imposing ﬁrst class constraints setting certain current componentsto constant values.In the rest of the paper, we assume that the WZNW group, G,is a connected real Lie group whose Lie algebra, G, is a non-compact real form of acomplex simple Lie algebra, Gc.

We shall ﬁrst uncover the Lie algebraic implicationsof the constraints being ﬁrst class, and also discuss a suﬃcient condition which may beused to ensure their conformal invariance. In particular, we shall see why the compactreal form is outside our framework.We then set up a gauged WZNW theory whichprovides a Lagrangean realization of the WZNW reduction, for the case of general ﬁrstclass constraints.

Finally, we shall describe the eﬀective ﬁeld theories resulting from thereduction in some detail in an important special case, namely when the left and rightKM currents are constrained for such subalgebras of G which are dual to each other withrespect to the Cartan-Killing form.2.1. First class and conformally invariant KM constraintsHere we analyze the general form of the KM constraints which will be used sub-sequently to reduce the WZNW theory.

The analysis applies to each current J and ˜Jseparately so we choose one of them, J say, for deﬁniteness. To ﬁx the conventions, weﬁrst note that the KM Poisson bracket reads{⟨u, J(x)⟩, ⟨v, J(y)⟩}|x0=y0 = ⟨[u, v], J(x)⟩δ(x1 −y1) + κ⟨u, v⟩δ′(x1 −y1),(2.1)where u and v are arbitrary generators of G and the inner product ⟨u , v⟩= Tr (u · v) isnormalized so that the long roots of Gc have length squared 2.

This normalization meansthat in terms of the adjoint representation one has ⟨u , v⟩=12gtr (adu · adv), where gis the dual Coxeter number. It is worth noting that ⟨u , v⟩is the usual matrix tracein the deﬁning, vector representation for the classical Lie algebras Al and Cl, and itis 12 × trace in the deﬁning representation for the Bl and Dl series.

We also wish topoint out that the KM Poisson bracket together with all the subsequent relations whichfollow from it hold in the same form both on the usual canonical phase space and on the11

space of the classical solutions of the theory. This is the advantage of using equal timePoisson brackets and spatial δ-functions even on the latter space, where J(x) dependson x = (x0, x1) only through x+ (see the footnote on page 7).The KM reduction we consider is deﬁned by requiring the constrained current to beof the following special form:J(x) = κM + j(x) ,withj(x) ∈Γ⊥,(2.2)where Γ is some linear subspace and M is some element of G. Equivalently, the constraintscan be given asφγ(x) = ⟨γ , J(x)⟩−κ⟨γ , M⟩= 0 ,∀γ ∈Γ .

(2.3)In words, our constraints set the current components corresponding to Γ to constantvalues.It is clear both from (2.2) and (2.3) that M can be shifted by an arbitraryelement from the space Γ⊥without changing the actual content of the constraints. Thisambiguity is unessential, since one can ﬁx M, for example, by requiring that it is fromsome given linear complement of Γ⊥in G, which can be chosen by convention.In our method we assume that the above system of constraints is ﬁrst class, andnow we analyze the content of this condition.

Immediately from (2.1), we have*{φα(x), φβ(y)} = φ[α,β](x)δ(x1 −y1) + ωM(α, β)δ(x1 −y1) + ⟨α, β⟩δ′(x1 −y1),(2.4)where the second term contains the restriction to Γ of the following anti-symmetric 2-formof G:ωM(u, v) ≡⟨M , [u , v]⟩,∀u , v ∈G . (2.5)It is evident from (2.4) that the constraints are ﬁrst class if, and only if, we have[α , β] ∈Γ,⟨α , β⟩= 0andωM(α , β) = 0,for∀α , β ∈Γ.

(2.6)This means that the linear subspace Γ has to be a subalgebra on which the Cartan-Killing form and ωM vanish. It is easy to see that the three conditions in (2.6) can beequivalently written as[Γ , Γ⊥] ⊂Γ⊥,Γ ⊂Γ⊥and[M , Γ] ⊂Γ⊥,(2.7)* For simplicity, we set κ to 1 in the rest of the paper, except in Chapter 5, where κoccurs in the formula of the Virasoro centre.12

respectively. Subalgebras Γ satisfying Γ ⊂Γ⊥exist in every real form of the complexsimple Lie algebras except the compact one, since for the compact real form the Cartan-Killing inner product is (negative) deﬁnite.We note that for a given Γ the third condition and the ambiguity in choosing M canbe concisely summarized by the (equivalent) statement thatM ∈[Γ , Γ]⊥/Γ⊥.

(2.8)The constraints deﬁned by the zero element of this factor-space are in a sense trivial.It is clear that, for a subalgebra Γ such that Γ ⊂Γ⊥, the above factor-space containsnon-zero elements if and only if [Γ, Γ] ̸= Γ. Actually this is always so because Γ ⊂Γ⊥implies that Γ is a solvable subalgebra of G. To prove this, we ﬁrst note that if Γ isnot solvable then, by Levi’s theorem [33], it contains a semi-simple subalgebra, in whichone can ﬁnd either an so(3, R) or an sl(2, R) subalgebra.

From this one sees that thereexists at least one generator λ of Γ for which the operator adλ is diagonalizable with realeigenvalues. It cannot be that all eigenvalues of adλ are 0 since G is a simple Lie algebra,and from this one gets that ⟨λ , λ⟩̸= 0, which contradicts Γ ⊂Γ⊥.

Therefore one canconclude that Γ is necessarily a solvable subalgebra of G.The second condition in (2.6) can be satisﬁed for example by assuming that everyγ ∈Γ is a nilpotent element of G. This is true in the concrete instances of the reductionstudied in Chapters 3 and 4. We note that in this case Γ is actually a nilpotent Liealgebra, by Engel’s theorem [33].However, the nilpotency of Γ is not necessary forsatisfying Γ ⊂Γ⊥.

In fact, a solvable but not nilpotent Γ can be found in Appendix A.The current components constrained in (2.3) are the inﬁnitesimal generators of theKM transformations corresponding to the subalgebra Γ, which act on the KM phasespace asJ(x) −→eai(x+)γi J(x) e−ai(x+)γi + (eai(x+)γi)′ e−ai(x+)γi ,(2.9)where the ai(x+) are parameter functions and there is a summation over some basisγi of Γ. Of course, the ﬁrst class conditions are equivalent to the statement that theconstraint surface, consisting of currents of the form (2.2), is left invariant by the abovetransformations.

From the point of view of the reduced theory, these transformationsare to be regarded as gauge transformations, which means that the reduced phase spacecan be identiﬁed as the space of gauge orbits in the constraint surface. Taking this intoaccount, we shall often refer to Γ as the gauge algebra of the reduction.13

We next discuss a suﬃcient condition for the conformal invariance of the constraints.We assume that M /∈Γ⊥from now on. The standard conformal symmetry generatedby the Sugawara Virasoro density LKM(x) is then broken by the constraints (2.3), sincethey set some component of the current, which has spin 1, to a non-zero constant.

Theidea is to circumvent this apparent violation of conformal invariance by changing thestandard action of the conformal group on the KM phase space to one which does leavethe constraint surface invariant. One can try to generate the new conformal action bychanging the usual KM Virasoro density to the new Virasoro densityLH(x) = LKM(x) −⟨H, J′(x)⟩,(2.10)where H is some element of G. The conformal action generated by LH(x) operates onthe KM phase space asδf,H J(x) ≡−Zdy1 f(y+) {LH(y) , J(x)}= f(x+)J′(x) + f ′(x+)J(x) + [H, J(x)]+ f ′′(x+)H ,(2.11)for any parameter function f(x+), corresponding to the conformal coordinate transfor-mation δf x+ = −f(x+).

In particular, j(x) in (2.2) transforms under this new conformalaction according toδf,H j(x) = f(x+)j′(x) + f ′′(x+)H + f ′(x+)j(x) + [H, j(x)] + ([H, M] + M), (2.12)and our condition is that this variation should be in Γ⊥, which means that this conformalaction preserves the constraint surface. From (2.12), one sees that this is equivalent tohaving the following relations:H ∈Γ⊥,[H, Γ⊥] ⊂Γ⊥and([H, M] + M) ∈Γ⊥.

(2.13)In conclusion, the existence of an operator H satisfying these relations is a suﬃcientcondition for the conformal invariance of the KM reduction obtained by imposing (2.3).The conditions in (2.13) are equivalent to LH(x) being a gauge invariant quantity, induc-ing a corresponding conformal action on the reduced phase space. Obviously, the secondrelation in (2.13) is equivalent to[H, Γ] ⊂Γ .

(2.14)An element H ∈G is called diagonalizable if the linear operator adH possesses acomplete set of eigenvectors in G. By the eigenspaces of adH, such an element deﬁnes a14

grading of G, and below we shall refer to a diagonalizable element as a grading operator ofG. In the examples we study later, conformal invariance will be ensured by the existenceof a grading operator subject to (2.13).If H is a grading operator satisfying (2.13) then it is always possible to shift M bysome element of Γ⊥(i.e., without changing the physics) so that the new M satisﬁes[H, M] = −M ,(2.15)instead of the last condition in (2.13).

It is also clear that if H is a grading operator thenone can take graded bases in Γ and Γ⊥, since these are invariant subspaces under adH.On re-inserting (2.15) into (2.12) it then follows that all components of j(x) are primaryﬁelds with respect to the conformal action generated by LH(x), with the exception ofthe H-component, which also survives the constraints according to the ﬁrst condition in(2.13).As an example, let us now consider some arbitrary grading operator H and denoteby Gm the eigensubspace corresponding to the eigenvalue m of adH. Then the gradedsubalgebra G≥n, which is deﬁned to be the direct sum of the subspaces Gm for all m ≥n,will qualify as a gauge algebra Γ for any n > 0 from the spectrum of adH.

In this caseΓ⊥= G>−n and the factor space [Γ, Γ]⊥/Γ⊥, which is the space of the allowed M’s, canbe represented as the direct sum of G−n and that graded subspace of G<−n which isorthogonal to [Γ, Γ]. It is easy to see that one obtains conformally invariant ﬁrst classconstraints by choosing M to be any graded element from this factor space.

Indeed, ifthe grade of M is −m then LH/m yields a Virasoro density weakly commuting with thecorresponding constraints.In summary, in this section we have seen that one can associate a ﬁrst class systemof KM constraints to any pair (Γ,M) subject to (2.6) by requiring the constrained currentto take the form (2.2), and that the conformal invariance of this system of constraintsis guaranteed if one can ﬁnd an operator H such that the triple (Γ,M,H) satisﬁes theconditions in (2.13).2.2. Lagrangean realization of the Hamiltonian reductionWe shall exhibit here a gauged WZNW theory providing the Lagrangean realizationof those Hamiltonian reductions of the WZNW theory which can be deﬁned by imposing15

ﬁrst class constraints of the type (2.3) on the KM currents J and ˜J of the theory. Itshould be noted that, in the rest of this chapter, we do not assume that the constraintsare conformally invariant.To deﬁne the WZNW reduction, we can choose left and right constraints completelyindependently.

We shall denote the pairs consisting of an appropriate subalgebra and aconstant matrix corresponding to the left and right constraints as (Γ, M) and (˜Γ, −˜M),respectively.The reduced theory is obtained by ﬁrst constraining the WZNW phasespace by settingφi = ⟨γi , J⟩−⟨γi , M⟩= 0,and˜φi = −⟨˜γi , ˜J⟩−⟨˜γi , ˜M⟩= 0,(2.16)where the γi and the ˜γi form bases of Γ and ˜Γ, respectively, and then factorizing theconstraint surface by the canonical transformations generated by these constraints. Onecan apply this reduction either to the usual canonical phase space or to the space ofsolutions of the classical ﬁeld equation.

These are equivalent procedures since the twospaces in question are isomorphic. For later purpose we note that the constraints generatethe following chiral gauge transformations on the space of solutions:g(x+, x−) −→eγ(x+) · g(x+, x−) · e−˜γ(x−) ,(2.17)where γ(x+) and ˜γ(x−) are arbitrary Γ and ˜Γ valued functions.For completeness, we wish to mention here how the above way of reducing theWZNW theory ﬁts into the general theory of Hamiltonian (symplectic) reduction ofsymmetries [34].

In general, the Hamiltonian reduction is obtained by setting the phasespace functions generating the symmetry transformations through Poisson bracket (inother words, the components of the momentum map) to some constant values.Thereduced phase space results by factorizing this constraint surface by the subgroup ofthe symmetry group respecting the constraints. The symmetry group we consider is theleft × right KM group generated by Γ × ˜Γ and our Hamiltonian reduction is special inthe sense that the full symmetry group preserves the constraints.

Of course, the latterfact is just a reformulation of the ﬁrst-classness of our constraints.We now come to the main point of the section, which is that the reduced WZNWtheory, deﬁned in the above by using the Hamiltonian picture, can be identiﬁed as thegauge invariant content of a corresponding gauged WZNW theory. This gauged WZNWinterpretation of the reduction was pointed out in the concrete case of the WZNW →standard Toda reduction in [13], and we below generalize that construction to the presentsituation.16

The gauged WZNW theory we are interested in is given by the following actionfunctional:I(g, A−, A+) ≡SWZ(g)+Zd2x⟨A−, ∂+gg−1 −M⟩+⟨A+, g−1∂−g −˜M⟩+ ⟨A−, gA+g−1⟩,(2.18)where the gauge ﬁelds A−(x) and A+(x) vary in Γ and ˜Γ, respectively. The main propertyof this action is that it is invariant under the following non-chiral gauge transformations:g →αg˜α−1;A−→αA−α−1 + α∂−α−1;A+ →˜αA+˜α−1 + (∂+˜α)˜α−1 ,(2.19a)whereα = eγ(x+,x−)and˜α = e˜γ(x+,x−) ,(2.19b)for any γ(x+, x−) ∈Γ and ˜γ(x+, x−) ∈˜Γ.

The proof of the invariance of (2.18) under(2.19) can proceed along the same lines as for the special case in [13]. In the proof onerewrites SWZ(αg˜α−1) by using the well-known Polyakov-Wiegmann identity [35], and inthis step one uses the fact that the WZNW action vanishes for ﬁelds in the subgroupsof G with Lie algebras Γ or ˜Γ.

This is an obvious consequence of the relations Γ ⊂Γ⊥and ˜Γ ⊂˜Γ⊥. The other crucial point is that the terms in (2.18) containing the constantmatrices M and˜M are separately invariant under (2.19).

It is easy to see that thisfollows from the third condition in (2.6).For example, under an inﬁnitesimal gaugetransformation belonging to α ≃1 + γ, the term ⟨A−, M⟩changes byδ ⟨A−, M⟩= −⟨∂−γ, M⟩+ ωM(γ, A−) ,(2.20)which is a total divergence since the second term vanishes, as both A−and γ are fromΓ.The Euler-Lagrange equation derived from (2.18) by varying g can be written equiv-alently as∂−(∂+gg−1 + gA+g−1) + [A−, ∂+gg−1 + gA+g−1] + ∂+A−= 0 ,(2.21a)or∂+(g−1∂−g + g−1A−g) −[A+, g−1∂−g + g−1A−g] + ∂−A+ = 0 ,(2.21b)and the ﬁeld equations obtained by varying A−and A+ are given by⟨γ , ∂+gg−1 + gA+g−1 −M⟩= 0,∀γ ∈Γ ,(2.21c)17

and⟨˜γ , g−1∂−g + g−1A−g −˜M⟩= 0,∀˜γ ∈˜Γ ,(2.21d)respectively. We now note that by making use of the gauge invariance, A+ and A−canbe set equal to zero simultaneously.

The important point for us is that, as is easy tosee, in the A± = 0 gauge one recovers from (2.21) both the ﬁeld equations (1.3) of theWZNW theory and the constraints (2.16). Furthermore, one sees that setting A± to zerois not a complete gauge ﬁxing, the residual gauge transformations are exactly the chiralgauge transformations of equation (2.17).The above arguments tell us that the space of gauge orbits in the space of classicalsolutions of the gauged WZNW theory (2.18) can be naturally identiﬁed with the reducedphase space belonging to the Hamiltonian reduction of the WZNW theory determined bythe ﬁrst class constraints (2.16).

It can be also shown that the Poisson bracket induced onthe reduced phase space by the Hamiltonian reduction is the same as the one determinedby the gauged WZNW action (2.18). In summary, we see that the gauged WZNW theory(2.18) provides a natural Lagrangean implementation of the WZNW reduction.2.3.

Eﬀective ﬁeld theories from left-right dual reductionsThe aim of this section is to describe the eﬀective ﬁeld equations and action func-tionals for an important class of the reduced WZNW theories. This class of theories isobtained by making the assumption that the left and right gauge algebras Γ and ˜Γ aredual to each other with respect to the Cartan-Killing form, which means that one canchoose bases γi ∈Γ and ˜γj ∈˜Γ so that⟨γi, ˜γj⟩= δij .

(2.22)This technical assumption allows for having a simple general algorithm for disentanglingthe constraints:φi = ⟨γi, ∂+g g−1 −M⟩= 0,and˜φi = ⟨˜γi, g−1∂−g −˜M⟩= 0,(2.23)which deﬁne the reduction. We shall comment on the physical meaning of the assumptionat the end of the section, here we only point out that it holds, e.g., if one chooses Γ and18

˜Γ to be the images of each other under a Cartan involution* of the underlying simple Liealgebra.For concreteness, let us consider the maximally non-compact real form which canbe deﬁned as the real span of a Chevalley basis Hi, E±α of the corresponding complexLie algebra Gc, and in the case of the classical series An, Bn, Cn and Dn is given bysl(n + 1, R), so(n, n + 1, R), sp(2n, R) and so(n, n, R), respectively.In this case theCartan involution is (−1) × transpose, operating on the Chevalley basis according toHi −→−HiE±α −→−E∓α . (2.24)It is obvious that ⟨v , vt⟩> 0 for any non-zero v ∈G and from this one sees that Γt isdual to Γ with respect to the Cartan-Killing form, i.e., (2.22) holds for ˜Γ = Γt.

It shouldalso be mentioned that there is a Cartan involution for every non-compact real form ofthe complex simple Lie algebras, as explained in detail in [36].Equation (2.22) implies that the left and right gauge algebras do not intersect, andthus we can consider a direct sum decomposition of G of the formG = Γ + B + ˜Γ ,(2.25a)where B is some linear subspace of G. Here B is in principle an arbitrary complementaryspace to (Γ + ˜Γ) in G, but one can always make the choiceB = (Γ + ˜Γ)⊥,(2.25b)which is natural in the sense that the Cartan-Killing form is non-degenerate on thisB. Choosing B according to (2.25b) is especially well-suited in the case of the parityinvariant eﬀective theories discussed at the end of the section.

We note that it mightalso be convenient if one can take the space B to be a subalgebra of G, but this is notnecessary for our arguments and is not always possible either.We can associate a ‘generalized Gauss decomposition’ of the group G to the directsum decomposition (2.25), which is the main tool of our analysis. By ‘Gauss decomposing’an element g ∈G according to (2.25), we mean writing it in the formg = a · b · c ,witha = eγ,b = eβandc = e˜γ,(2.26)* A Cartan involution σ of the simple Lie algebra G is an automorphism for whichσ2 = 1 and ⟨v, σ(v)⟩< 0 for any non-zero element v of G.19

where γ, β and ˜γ are from the respective subspaces in (2.25).There is a neighbourhood of the identity in G consisting of elements which allow aunique decomposition of this sort, and in this neighbourhood the pieces a, b and c canbe extracted from g by algebraic operations. (Actually it is also possible to deﬁne b asa product of exponentials corresponding to subspaces of B, and we shall make use ofthis freedom later, in Chapter 4.) We make the assumption that every G-valued ﬁeld weencounter is decomposable as g in (2.26).

It is easily seen that in this ‘Gauss decompos-able sector’ the components of b(x+, x−) provide a complete set of gauge invariant localﬁelds, which are the local ﬁelds of the reduced theory we are after. Below we explainhow to solve the constraints (2.23) in the Gauss decomposable sector of the WZNWtheory.

More exactly, for our method to work, we restrict ourselves to considering thoseﬁelds which vary in such a Gauss decomposable neighbourhood of the identity where thematrixVij(b) = ⟨γi, b˜γjb−1⟩(2.27)is invertible. Due to the assumptions, the analysis given in the following yields a localdescription of the reduced theories.

It is clear that for a global description one shoulduse patches on G obtained by multiplying out the Gauss decomposable neighbourhoodof the identity, but we do not deal with this issue here.First we derive the ﬁeld equation of the reduced theory by implementing the con-straints directly in the WZNW ﬁeld equation ∂−(∂+gg−1) = 0. (This is allowed since theWZNW dynamics leaves the constraint surface invariant, i.e., the WZNW Hamiltonianweakly commutes with the constraints.) By inserting the Gauss decomposition of g into(2.23) and making use of the constraints being ﬁrst class, the constraint equations canbe rewritten as⟨γi, ∂+bb−1 + b(∂+cc−1)b−1 −M⟩= 0 ,⟨˜γi, b−1∂−b + b−1(a−1∂−a)b −˜M⟩= 0 .

(2.28)With the help of the inverse of Vij(b) in (2.27), one can solve these equations for ∂+cc−1and a−1∂−a in terms of b,∂+cc−1 = b−1T(b)b,anda−1∂−a = b ˜T(b)b−1,(2.29a)whereT(b) =XijV −1ij (b)⟨γj, M −∂+bb−1⟩b˜γib−1,˜T(b) =XijV −1ij (b)⟨˜γi, ˜M −b−1∂−b⟩b−1γjb . (2.29b)20

It is easy to obtain the eﬀective ﬁeld equation for the ﬁeld b(x+, x−) by using this explicitform of the constraints. This can be achieved for example by noting that, by applyingthe operator Ada−1 to equation (1.9) (i.e., by conjugating it by a−1) the WZNW ﬁeldequation can be written in the form[∂+ −A+ , ∂−−A−] = 0(2.30)withA+ = ∂+b b−1 + b(∂+cc−1)b−1andA−= −a−1∂−a .

(2.31)Thus, by inserting the constraints (2.29) into the above form of the WZNW equation,we see that the ﬁeld equation of the reduced theory is the zero curvature condition ofthe following Lax potential:A+(b) = ∂+b b−1 + T(b)andA−(b) = −b ˜T(b)b−1 . (2.32)More explicitly, the eﬀective ﬁeld equation reads∂−(∂+bb−1) + [b ˜T(b)b−1, T(b)] + ∂−T(b) + b(∂+ ˜T(b))b−1 = 0.

(2.33)The expression on the left-hand-side of (2.33) in general varies in the full space G, butnot all the components represent independent equations. The number of the independentequations is the number of the independent components of the WZNW ﬁeld equationminus the number of the constraints in (2.23), since the constraints automatically implythe corresponding components of the WZNW equation.

Thus there are exactly as manyindependent equations in (2.33) as the number of the reduced degrees of freedom. Infact, the independent ﬁeld equations can be obtained by taking the Cartan-Killing innerproduct of (2.33) with a basis of the linear space B in (2.25), and the inner product of(2.33) with the γi and the ˜γi vanishes as a consequence of the constraints in (2.23) togetherwith the independent ﬁeld equations.

To see this one ﬁrst recalls that the left-hand-side of(2.33) is, upon imposing the constaints, equivalent to a−1(∂−J)a. Thus the inner productof this with Γ, and similarly that of c(∂+ ˜J)c−1 with ˜Γ, vanishes as a consequence of theconstraints.

From this, by using the identity a−1(∂−J)a = −bc(∂+ ˜J)c−1b−1, one canconclude that the inner product of a−1(∂−J)a with ˜Γ also vanishes as a consequence ofthe constraints and the independent ﬁeld equations.At this point we would like to mention certain special cases when the above equationssimplify. First we note that if one has[B , Γ] ⊂Γand[B , ˜Γ] ⊂˜Γ ,(2.34)21

thenT(b) = M −π˜Γ(∂+bb−1)and˜T(b) = ˜M −πΓ(b−1∂−b) ,(2.35)where we introduced the operatorsπΓ =Xi|γi⟩⟨˜γi|andπ˜Γ =Xi|˜γi⟩⟨γi| ,(2.36)which project onto the spaces Γ and ˜Γ, and assumed that M ∈˜Γ and ˜M ∈Γ. (The latterassumption can be done without loss of generality due to the duality condition (2.22)).One obtains (2.35) from (2.29) by taking into account that in this case Vij(b) in (2.27) isthe matrix of the operator Adb acting on ˜Γ, and thus the inverse is given by Adb−1.

Thenicest possible situation occurs when B = (Γ + ˜Γ)⊥is a subalgebra of G and also satisﬁes(2.34). In this case one simply has T = M and ˜T = ˜M and thus (2.33) simpliﬁes to∂−(∂+bb−1) + [b ˜Mb−1 , M] = 0 .

(2.37)The derivative term is now an element of B and by combining the above assumptions withthe ﬁrst class conditions [M, Γ] ⊂Γ⊥and [ ˜M, ˜Γ] ⊂˜Γ⊥one sees that the commutatorterm in (2.37) also varies in B, which ensures the consistency of this equation.The eﬀective ﬁeld equation (2.33) is in general a non-linear equation for the ﬁeldb(x+, x−), and we can give a procedure which can in principle be used for producing itsgeneral solution. We are going to do this by making use of the fact that the space ofsolutions of the reduced theory is the space of the constrained WZNW solutions factorizedby the chiral gauge transformations, according to equation (2.17).

Thus the idea is toﬁnd the general solution of the eﬀective ﬁeld equation by ﬁrst parametrizing, in terms ofarbitrary chiral functions, those WZNW solutions which satisfy the constraints (2.23),and then extracting the b-part of those WZNW solutions by algebraic operations. Inother words, we propose to derive the general solution of (2.33) by looking at the originof this equation, instead of its explicit form.To be more concrete, one can start the construction of the general solution byﬁrst Gauss-decomposing the chiral factors of the general WZNW solution g(x+, x−) =gL(x+) · gR(x−) asgL(x+) = aL(x+) · bL(x+) · cL(x+),gR(x−) = aR(x−) · bR(x−) · cR(x−).

(2.38)Then the constraint equations (2.23) become∂+cLc−1L= b−1L T(bL)bLanda−1R ∂−aR = bR ˜T(bR)b−1R . (2.39)22

In addition to the the purely algebraic problems of computing the quantities T and ˜T andextracting b from g = gL · gR = a · b · c, these ﬁrst order systems of ordinary diﬀerentialequations are all one has to solve to produce the general solution of the eﬀective ﬁeldequation.If this can be done by quadrature then the eﬀective ﬁeld equation is alsointegrable by quadrature. In general, one can proceed by trying to solve (2.39) for thefunctions cL(x+) and aR(x−) in terms of the arbitrary ‘input functions’ bL(x+) andbR(x−).

Clearly, this involves only a ﬁnite number of integrations whenever the gaugealgebras Γ and ˜Γ consist of nilpotent elements of G. Thus in this case (2.33) is exactlyintegrable, i.e., its general solution can be obtained by quadrature.We note that in concrete cases some other choice of input functions, instead of thechiral b’s, might prove more convenient for ﬁnding the general solutions of the systemsof ﬁrst order equations on gL and gR given in (2.39) (see for instance the derivation ofthe general solution of the Liouville equation given in [12]).It is natural to ask for the action functional underlying the eﬀective ﬁeld theoryobtained by imposing the constraints (2.23) on the WZNW theory. In fact, the eﬀectiveaction is given by the following formula:Ieﬀ(b) = SWZ(b) −Zd2x ⟨b ˜T(b)b−1 , T(b)⟩.

(2.40)One can derive the following condition for the extremum of this action:⟨δbb−1, ∂−(∂+bb−1) + [b ˜T(b)b−1, T(b)] + ∂−T(b) + b(∂+ ˜T(b))b−1⟩= 0. (2.41)It is straightforward to compute this, the only thing to remember is that the objectsb ˜Tb−1 and b−1Tb introduced in (2.29) vary in the gauge algebras Γ and ˜Γ.

The arbitraryvariation of b(x) is determined by the arbitrary variation of β(x) ∈B, according tob(x) = eβ(x), and thus we see from (2.41) that the Euler-Lagrange equation of the action(2.40) yields exactly the independent components of the eﬀective ﬁeld equation (2.33),which we obtained previously by imposing the constraints directly in the WZNW ﬁeldequation.The eﬀective action given above can be derived from the gauged WZNW actionI(g, A−, A+) given in (2.18), by eliminating the gauge ﬁelds A± by means of their Euler-Lagrange equations (2.21c-d). By using the Gauss decomposition, these Euler-Lagrangeequations become equivalent to the relationsa−1D−a = b ˜T(b)b−1 ,andcD+c−1 = −b−1T(b)b ,(2.42)23

where the quantities T(b) and ˜T(b) are given by the expressions in (2.29b) and D± denotesthe gauge covariant derivatives, D± = ∂± ∓A±. Now we show that Ieﬀ(b) in (2.40) canindeed be obtained by substituting the solution of (2.42) for A± back into I(g, A−, A+)with g = abc.

To this ﬁrst we rewrite I(abc, A−, A+) by using the Polyakov-Wiegmannidentity [35] asI(abc , A−, A+) = SWZ(b) −Zd2x⟨a−1D−a , b(cD+c−1)b−1⟩+ ⟨b−1∂−b , cD+c−1⟩−⟨∂+bb−1 , a−1D−a⟩+ ⟨A−, M⟩+ ⟨A+, ˜M⟩. (2.43)This equation can be regarded as the gauge covariant form of the Polyakov-Wiegmannidentity, and all but the last two terms are manifestly gauge invariant.

The eﬀectiveaction (2.40) is derived from (2.43) together with (2.42) by noting, for example, that⟨∂−aa−1 , M⟩is a total derivative, which follows from the facts that a(x) ∈eΓ andM ∈[Γ , Γ]⊥, by (2.8).Above we have used the ﬁeld equations to eliminate the gauge ﬁelds from the gaugedWZNW action (2.18) on the ground that A−and A+ are not dynamical ﬁelds, but‘Lagrange multiplier ﬁelds’ implementing the constraints. However, it should be notedthat without further assumptions the Euler-Lagrange equation of the action resultingfrom (2.18) by means of this elimination procedure does not always give the eﬀectiveﬁeld equation, which can always be obtained directly from the WZNW ﬁeld equation.One can see this on an example in which one imposes constraints only on one of the chiralsectors of the WZNW theory.

From this point of view, the role of our assumption on theduality of the left and right gauge algebras is that it guarantees that the eﬀective actionunderlying the eﬀective ﬁeld equation can be derived from I(g, A−, A+) in the abovemanner. To end this discussion, we note that for g = abc the non-degeneracy of Vij(b)in (2.27) is equivalent to the non-degeneracy of the quadratic expression ⟨A−, gA+g−1⟩in the components of A−= Ai−γi and A+ = Ai+˜γi.

This quadratic term enters into thegauged WZNW action given by (2.18), and its non-degeneracy is clearly important inthe quantum theory, which we consider in Chapter 5.We mentioned at the beginning of the section that, considering a maximally non-compact G, one can make sure that the duality assumption expressed by (2.22) holdsby choosing Γ and ˜Γ to be the transposes of each other. Here we point out that thisparticular left-right related choice of the gauge algebras can also be used to ensure theparity invariance of the eﬀective ﬁeld theory.

To this ﬁrst we notice that, in the case of amaximally non-compact connected Lie group G, the WZNW action SWZ(g) is invariant24

under any of the following two ‘parity transformations’ g −→Pg:(P1g)(x0, x1) ≡gt(x0, −x1) ,and(P2g)(x0, x1) ≡g−1(x0, −x1). (2.44)If one chooses ˜Γ = Γt and˜M = M t to deﬁne the WZNW reduction then the paritytransformation P1 simply interchanges the left and right constraints, φ and ˜φ in (2.23),and thus the corresponding eﬀective ﬁeld theory is invariant under the parity P1.

Thespace B = (Γ + ˜Γ)⊥, i.e., the choice in (2.25b), is invariant under the transpose in thiscase, and thus the gauge invariant ﬁeld b transforms in the same way under P1 as gdoes in (2.44). Of course, the parity invariance can also be seen on the level of thegauged action I(g, A−, A+).

Namely, I(g, A−, A+) is invariant under P1 if one extendsthe deﬁnition in (2.44) to include the following parity transformation of the gauge ﬁelds:(P1A±)(x0, x1) ≡At∓(x0, −x1) . (2.45)The P1-invariant reduction procedure does not preserve the parity symmetry P2, but it ispossible to consider reductions preserving just P2 instead of P1.

In fact, such reductionscan be obtained by taking ˜Γ = Γ and ˜M = M.Finally, it is obvious that to construct parity invariant WZNW reductions in general,for some arbitrary but non-compact real form G of the complex simple Lie algebras, onecan use −σ instead of the transpose, where σ is a Cartan involution of G.25

3. Polynomiality in KM reductions and the WGS -algebrasIn the previous chapter we described the conditions for (2.2) deﬁning ﬁrst classconstraints and for LH(J) in (2.10) being a gauge invariant quantity on this constraintsurface.

It is clear that the KM Poisson brackets of the gauge invariant diﬀerential poly-nomials of the current always close on such polynomials and δ-distributions. The algebraof the gauge invariant diﬀerential polynomials is of special interest in the conformally in-variant case when it is a polynomial extension of the Virasoro algebra.

In Section 3.1 weshall give an additional condition on the triple (Γ, M, H) which allows one to constructout of the current in (2.2) a complete set of gauge invariant diﬀerential polynomials bymeans of a diﬀerential polynomial gauge ﬁxing algorithm. We call the KM reductionpolynomial if such a polynomial gauge ﬁxing algorithm is available, and also call thecorresponding gauges Drinfeld-Sokolov (DS) gauges, since our construction is a general-ization of the one given in [5].

The KM Poisson bracket algebra of the gauge invariantdiﬀerential polynomials becomes the Dirac bracket algebra of the current components inthe DS gauges, which we consider in Section 3.2. The extended conformal algebra WGSmentioned in the Introduction is especially interesting in that its primary ﬁeld basis ismanifest and given by the sl(2) structure, as we shall see in Section 3.3.

One of our mainresults is that we shall ﬁnd here ﬁrst class KM constraints underlying this algebra, suchthat they satisfy our suﬃcient condition for polynomiality. Thus we can represent WGSas a KM Poisson bracket algebra of gauge invariant diﬀerential polynomials, which inprinciple allows for its quantization through the KM representation theory.

The impor-tance of the WGS -algebras is clearly demonstrated by the result of Section 3.4, where weshow that the W ln-algebras of [26] can be interpreted as further reductions of particularWGS -algebras. This makes it possible to exhibit primary ﬁelds for the W ln-algebras andto describe their structure in detail in terms of the corresponding WGS -algebras, which isthe subject of [37].3.1.

A suﬃcient condition for polynomialityLet us suppose that (Γ, M, H) satisfy the previously given conditions, (2.6) and(2.13), forJ(x) = M + j(x) ,j(x) ∈Γ⊥(3.1)26

describing the constraint surface of conformally invariant ﬁrst class constraints, where His a grading operator and M is subject to[H, M] = −M ,M /∈Γ⊥. (3.2)Then, as we shall show, the following two additional conditions:Γ ∩KM = {0} ,whereKM = Ker(adM) ,(3.3)andΓ⊥⊂G>−1 ,(3.4a)allow for establishing a diﬀerential polynomial gauge ﬁxing algorithm whereby one canconstruct out of J(x) in (3.1) a complete set of gauge invariant diﬀerential polynomials.Before proving this result, we discuss some consequences of the conditions, whichwe shall need later.

In the present situation Γ, Γ⊥and G are graded by the eigenvaluesof adH, and ﬁrst we note that (3.4a) is equivalent toG≥1 ⊂Γ . (3.4b)Indeed, this follows from the fact that the spaces Gh and G−h are dual to each other withrespect to the Cartan-Killing form, which is a consequence of its non-degeneracy andinvariance under adH.

Of course, here and below the grading is the one deﬁned by H,and we note that G±1 are non-trivial because of (3.2). The condition given by (3.4a) playsa technical role in our considerations, but perhaps it can be argued for also physically,on the basis that it ensures that the conformal weights of the primary ﬁeld componentsof j(x) in (3.1) are non-negative with respect to LH (2.10).

Second, let us observe thatin our situation M satisfying (3.2) is uniquely determined, that is, there is no possibilityof shifting it by elements from Γ⊥, simply because there are no grade −1 elements in Γ⊥,on account of (3.4a). Equation (3.3) means that the operator adM maps Γ into Γ⊥in aninjective manner, and for this reason we refer to (3.3) as the non-degeneracy condition.Combining the non-degeneracy condition with (3.2), (3.4a) and (2.7) we see that ourgauge algebra Γ can contain only positive grades:Γ ⊂G>0 .

(3.5)This implies that every γ ∈Γ is represented by a nilpotent operator in any ﬁnite dimen-sional representation of G, and thatG≥0 ⊂Γ⊥. (3.6)27

It follows from (3.2) that [H, KM] ⊂KM, which is telling us that KM is also graded, andwe see from (3.3) and (3.4b) thatKM ⊂G<1 . (3.7)Finally, we wish to establish a certain relationship between the dimensions of G and KM.For this purpose we consider an arbitrary complementary space TM to KM, deﬁning alinear direct sum decompositionG = KM + TM .

(3.8)It is easy to see that for the 2-form ωM we have ωM(KM, G) = 0, and the restriction ofωM to TM is a symplectic form, in other words:ωM(TM, TM)isnon−degenerate . (3.9)(We note in passing that TM can be identiﬁed with the tangent space at M to thecoadjoint orbit of G through M, and in this picture ωM becomes the Kirillov-Kostantsymplectic form of the orbit [34].) The non-degeneracy condition (3.3) says that one canchoose the space TM in (3.8) in such a way that Γ ⊂TM.

One then obtains the inequalitydim(Γ) ≤12dim(TM) = 12dim(G) −dim(KM),(3.10)where the factor 12 arises since ωM is a symplectic form on TM, which vanishes, by (2.6),on the subspace Γ ⊂TM.After the above clariﬁcation of the meaning of conditions (3.3) and (3.4), we nowwish to show that they indeed allow for exhibiting a complete set of gauge invariantdiﬀerential polynomials among the gauge invariant functions. Generalizing the argumentsof [5,13,15], this will be achieved by demonstrating that an arbitrary current J(x) subjectto (3.1) can be brought to a certain normal form by a unique gauge transformation whichdepends on J(x) in a diﬀerential polynomial way.A normal form suitable for this purpose can be associated to any graded subspaceΘ ⊂G which is dual to Γ with respect to the 2-form ωM.

Given such a space Θ, it ispossible to choose bases γih and θjk in Γ and Θ respectively such thatωM(γlh, θik) = δilδhk,(3.11)where the subscript h on γlh denotes the grade, and the indices i and l denote theadditional labels which are necessary to specify the base vectors at ﬁxed grade. It is28

to be noted that, by deﬁnition, the subsript k on elements θjk ∈Θ does not denote thegrade, which is (1 −k). The normal (or reduced) form corresponding to Θ is given by thefollowing equation:Jred(x) = M + jred(x)wherejred(x) ∈Γ⊥∩Θ⊥.

(3.12)In other words, the set of reduced currents is obtained by supplementing the ﬁrst classconstraints of equation (2.3) by the gauge ﬁxing conditionχθ(x) = ⟨J(x), θ⟩−⟨M, θ⟩= 0 ,∀θ ∈Θ . (3.13)We call a gauge which can be obtained in the above manner a Drinfeld-Sokolov (DS)gauge.

It is not hard to see that the space V = Γ⊥∩Θ⊥is a graded subspace of Γ⊥which is disjoint from the image of Γ under the operator adM and is in fact complementaryto the image, i.e., one hasΓ⊥= [M, Γ] + V . (3.14)It also follows from the non-degeneracy condition (3.3) that any graded complement Vin (3.14) can be obtained in the above manner, by means of using some Θ.

Thus it ispossible to deﬁne the DS normal form of the current directly in terms of a complementaryspace V as well, as has been done in special cases in [5,13,18].As the ﬁrst step in proving that any current in (3.1) is gauge equivalent to one inthe DS gauge, let us consider the gauge transformation by gh(x+) = exp[Pl alh(x+)γlh]for some ﬁxed grade h. Suppressing the summation over l, it can be written as*j(x) →jgh(x) = eah·γh(j(x) + M)e−ah·γh + (eah·γh)′e−ah·γh −M . (3.15)Taking the inner product of this equation with the basis vectors θik in (3.11) for all k ≤h,we see that there is no contribution from the derivative term.

We also see that the onlycontribution fromeah·γhj(x)e−ah·γh = j(x) + [ah(x+) · γh, j(x)] + . .

. (3.16)* Throughout the chapter, all equations involving gauge transformations, Poissonbrackets, etc., are to be evaluated by using a ﬁxed time, since they are all consequencesof equation (2.1).

By this convention, they are valid both on the canonical phase spaceand on the chiral KM phase space belonging to space of solutions of the theory.29

is the one coming from the ﬁrst term, since all commutators containing the elements γlhdrop out from the inner product in question as a consequence of the following crucialrelation:[γlh, θik] ∈Γ,fork ≤h,(3.17)which follows from (3.4b) by noting that the grade of this commutator, (1 + h −k), is atleast 1 for k ≤h. Taking these into account, and computing the contribution from thosetwo terms in jgh(x) which contain M by using (3.11), we obtain⟨θik, jgh(x)⟩= ⟨θik, j(x)⟩−aih(x+)δhk,for allk ≤h.

(3.18)We see from this equation that⟨θik, j(x)⟩= 0⇐⇒⟨θik, jgh(x)⟩= 0 ,fork < h ,(3.19)andaih(x+) = ⟨θih, j(x)⟩⇒⟨θih, jgh(x)⟩= 0 ,fork = h.(3.20)These last two equations tell us that if the gauge-ﬁxing condition ⟨θik, j(x)⟩= 0 is satisﬁedfor all k < h then we can ensure that the same condition holds for jgh(x) for the extendedrange of indices k ≤h, by choosing aih(x+) to be ⟨θih, j(x)⟩. From this it is easy to seethat the DS gauge (3.13) can be reached by an iterative process of gauge transformations,and the gauge-parameters aih(x+) are unique polynomials in the current at each stage ofthe iteration.In more detail, let us write the general element g(a(x+)) ∈eΓ of the gauge group asa product in order of descending grades, i.e., asg(a(x+)) = ghn · ghn−1 · · ·gh1,withghi(x+) = eahi(x+)·γhi ,(3.21a)wherehn > hn−1 > .

. .

> h1(3.21b)is the list of grades occurring in Γ. Let us then insert this expression intoj →jg = g(j + M)g−1 + g′g−1 −M ,(3.22a)and consider the conditionjg(x) = jred(x) ,(3.22b)with jred(x) in (3.12), as an equation for the gauge-parameters ah(x+).

One sees fromthe above considerations that this equation is uniquely soluble for the components of30

the ah(x+) and the solution is a diﬀerential polynomial in j(x). This implies that thecomponents of jred(x) can also be uniquely computed from (3.22), and the solution yieldsa complete set of gauge invariant diﬀerential polynomials of j(x), which establishes therequired result.

The above iterative procedure is in fact a convenient tool for computingthe gauge invariant diﬀerential polynomials in practice [15]. We remark that, of course,any unique gauge ﬁxing can be used to deﬁne gauge invariant quantities, but they are ingeneral not polynomial, not even local in j(x).We also wish to note that an arbitrary linear subspace of G which is dual to V in(3.14) with respect to the Cartan-Killing form can be used in a natural way as the spaceof parameters for describing those current dependent KM transformations which preservethe DS gauge.

In fact, it is possible to give an algorithm which computes the W-algebraand its action on the other ﬁelds of the corresponding constrained WZNW theory byﬁnding the gauge preserving KM transformations implementing the W-transformations.This algorithm presupposes the existence of such gauge invariant diﬀerential polynomialswhich reduce to the current components in the DS gauge, which is ensured by the abovegauge ﬁxing algorithm, but it works without actually computing them. This issue istreated in detail in [13,18] in special cases, but the results given there apply also to thegeneral situation investigated in the above.3.2.

The polynomiality of the Dirac bracketIt follows from the polynomiality of the gauge ﬁxing that the components of thegauge ﬁxed current jred in (3.12) generate a diﬀerential polynomial algebra under Diracbracket. In our proof of the polynomiality we actually only used that the graded subspaceΘ of G is dual to the graded gauge algebra Γ with respect to ωM and satisﬁes the condition([Θ , Γ])≥1 ⊂Γ ,(3.23)which is equivalent to the existence of the bases γlh and θik satisfying (3.11) and (3.17).We have seen that this condition follows from (3.3) and (3.4), but it should be noted thatit is a more general condition, since the converse is not true, as is shown by an exampleat the end of this section.Below we wish to give a direct proof for the polynomiality of the Dirac bracket31

algebra belonging to the second class constraints:cτ(x) = ⟨τ , J(x) −M⟩= 0whereτ ∈{γlh} ∪{θik} . (3.24)The proof will shed a new light on the polynomiality condition.

We note that for certainpurposes second class constraints might be more natural to use than ﬁrst class ones sincein the second class formalism one directly deals with the physical ﬁelds. For example, theWGS -algebra mentioned in the Introduction is very natural from the second class pointof view and can be realized by starting with a number of diﬀerent ﬁrst class systems ofconstraints, as we shall see in the next section.We ﬁrst recall that, by deﬁnition, the Dirac bracket algebra of the reduced currentsis{jured(x),jvred(y)}∗= {jured(x), jvred(y)}−XµνZdz1dw1{jured(x), cµ(z)}∆µν(z, w){cν(w), jvred(y)} ,(3.25)where, for any u ∈G, jured(x) = ⟨u, jred(x)⟩is to be substituted by ⟨u, J(x) −M⟩underthe KM Poisson bracket, and ∆µν(z, w) is the inverse of the kernelDµν(z, w) = {cµ(z), cν(w)} ,(3.26)in the sense that (on the constraint surface)XνZdx1∆µν(z, x)Dνσ(x, w) = δµσδ(z1 −w1).

(3.27)To establish the polynomiality of the Dirac bracket, it is useful to consider the matrixdiﬀerential operator Dµν(z) deﬁned by the kernel Dµν(z, w) in the usual way, i.e.,XνDµν(z)fν(z) =XνZdw1Dµν(z, w)fν(w) ,(3.28)for a vector of smooth functions fν(z), which are periodic in z1. From the structure ofthe constraints in (3.24), cτ = (φγ, χθ), one sees that Dµν(z) is a ﬁrst order diﬀerentialoperator possessing the following block structureDµν =Dγ˜γDγθD˜θ˜γD˜θθ=0E−E†F,(3.29)where E† is the formal Hermitian conjugate of the matrix E, (E†)θγ = (Eγθ)†.

It is clearthat the Dirac bracket in (3.25) is a diﬀerential polynomial in jred(x) and δ(x1 −y1)32

whenever the inverse operator D−1(z), whose kernel is ∆µν(z, w), is a diﬀerential operatorwhose coeﬃcients are diﬀerential polynomials in jred(z). On the other hand, we see from(3.29) that the operator D is invertible if and only if its block E is invertible, and in thatcase the inverse takes the form(D−1)µν =(E†)−1FE−1−(E†)−1E−10.

(3.30)Since E(z) and F(z) are polynomial (even linear) in jred(z) and in ∂z and the inverseof F(z) does not occur in D−1(z), it follows that D−1(z) is a polynomial diﬀerentialoperator if and only if E−1(z) is a polynomial diﬀerential operator.To show that E−1 exists and is a polynomial diﬀerential operator we note that interms of the basis of (Γ + Θ) in (3.24) the matrix E is given explicitly by the followingformula:Eγmh ,θnk (z) = δhkδmn + ⟨[γmh , θnk], jred(z)⟩+ ⟨γmh , θnk⟩∂z . (3.31)The crucial point is that, by the grading and the property in (3.17), we haveEγmh ,θnk (z) = δhkδnm ,fork ≤h .

(3.32)The matrix E has a block structure labelled by the (block) row and (block) columnindices h and k, respectively, and (3.32) means that the blocks in the diagonal of E areunit matrices and the blocks below the diagonal vanish. In other words, E is of the formE = 1 + ε, where ε is a strictly upper triangular matrix.

It is clear that such a matrixdiﬀerential operator is polynomially invertible, namely by a ﬁnite series of the formE−1 = 1 −ε + ε2 + . .

. + (−1)NεN ,(εN+1 = 0),(3.33)which ﬁnishes our proof of the polynomiality of the Dirac bracket in (3.25).

One can usethe arguments in the above proof to set up an algorithm for actually computing the Diracbracket. The proof also shows that the polynomiality of the Dirac bracket is guaranteedwhenever E is of the form (1+ε) with ε being nilpotent as a matrix.

In our case this wasensured by a special grading assumption, and it appears an interesting question whetherpolynomial reductions can be obtained at all without using some grading structure.The zero block occurs in D−1 in (3.30) because the second class constraints originatefrom the gauge ﬁxing of ﬁrst class ones. We note that the presence of this zero blockimplies that the Dirac brackets of the gauge invariant quantities coincide with theiroriginal Poisson brackets, namely one sees this from the formula of the Dirac bracket by33

keeping in mind that the gauge invariant quantities weakly commute with the ﬁrst classconstraints.Finally, we want to show that condition (3.23) is weaker than (3.3-4). This is bestseen by considering an example.

To this let now G be the maximally non-compact realform of a complex simple Lie algebra. If {M−, M0, M+} is the principal sl(2) embeddingin G, with commutation rules as in (3.34) below, we simply choose the one-dimensionalgauge algebra Γ ≡{M+} and take M ≡M−.

The ωM-dual to M+ can be taken tobe θ = M0, and then (3.23) holds. To show that conditions (3.3-4) cannot be satisﬁed,we prove that a grading operator H for which [H, M−] = −M−and GH≥1 ⊂Γ, does notexist.

First of all, [H, M−] = −M−and ⟨M−, M+⟩̸= 0 imply [H, M+] = M+, and thusΓH≥1 = {M+}. Furthermore, writing H = (M0 + ∆), we ﬁnd from [H, M±] = ±M± that∆must b e an sl(2) singlet in the adjoint of G. However, in the case of the principalsl(2) embedding, there is no such singlet in the adjoint, and hence H = M0.

But thenthe condition GM0≥1 ⊂Γ is not fulﬁlled.3.3. First class constraints for the WGS -algebrasLet S = {M−, M0 , M+} be an sl(2) subalgebra of the simple Lie algebra G:[M0, M±] = ±M± ,[M+, M−] = 2M0 .

(3.34)We argued in the Introduction that it is natural to associate an extended conformalalgebra, denoted as WGS , to any such sl(2) embedding [16,18]. Namely, we deﬁned theWGS -algebra to be the Dirac bracket algebra generated by the components of the con-strained KM current of the the following special form:Jred(x) = M−+ jred(x) ,withjred(x) ∈Ker(adM+) ,(3.35)which means that jred(x) is a linear combination of the sl(2) highest weight states in theadjoint of G. This deﬁnition is indeed natural in the sense that the conformal propertiesare manifest, since, as we shall see below, with the exception of the M+-componentthe spin s component of jred(x) turns out to be a primary ﬁeld of conformal weight(s + 1) with respect to LM0.

Before showing this, we shall construct here ﬁrst class KMconstraints underlying the WGS -algebra, which will be used in Chapter 4 to constructgeneralized Toda theories which realize the WGS -algebras as their chiral algebras. We34

expect the WGS -algebras to play an important organizing role in describing the (primaryﬁeld content of) conformally invariant KM reductions in general, and shall give argumentsin favour of this idea later.We wish to ﬁnd a gauge algebra Γ for which the triple (Γ, H = M0, M = M−)satisﬁes our suﬃcient conditions for polynomiality and (3.35) represents a DS gauge forthe corresponding conformally invariant ﬁrst class constraints. We start by noticing thatthe dimension of such a Γ has to satisfy the relationdim Ker(adM+) = dim WGS = dim G −2dim Γ .

(3.36)From this, since the kernels of adM± are of equal dimension, we obtain thatdim Γ = 12dim G −12dim Ker(adM−) ,(3.37)which means by (3.10) that we are looking for a Γ of maximal dimension. By the repre-sentation theory of sl(2), the above equality is equivalent todim Γ = dim G≥1 + 12dim G 12 ,(3.38)where the grading is by the, in general half-integral, eigenvalues of adM0.

We also know,(3.4b) and (3.5), that for our purpose we have to choose the graded Lie subalgebra Γ ofG in such a way that G≥1 ⊂Γ ⊂G>0. Observe that the non-degeneracy condition (3.3)is automatically satisﬁed for any such Γ since in the present case Ker(adM−) ⊂G≤0, andM0 ∈Γ⊥is also ensured, which guarantees the conformal invariance, see (2.13).It is obvious from the above that in the special case of an integral sl(2) subalgebra,for which G 12 is empty, one can simply takeΓ = G≥1 .

(3.39)For grading reasons,ωM−(G≥1, G≥1) = 0(3.40)holds, and thus one indeed obtains ﬁrst class constraints in this way.One sees from (3.38) that for ﬁnding the gauge algebra in the non-trivial case of ahalf-integral sl(2) subalgebra, one should somehow add half of G 12 to G≥1, in order tohave the correct dimension. The key observation for deﬁning the required halving of G 12consists in noticing that the restriction of the 2-form ωM−to G 12 is non-degenerate.

This35

can be seen as a consequence of (3.9), but is also easy to verify directly. By the wellknown Darboux normal form of symplectic forms [34], there exists a (non-unique) directsum decompositionG 12 = P 12 + Q 12(3.41)such that ωM−vanishes on the subspaces P 12 and Q 12 separately.

The spaces P 12 andQ 12 , which are the analogues of the usual momentum and coordinate subspaces of thephase space in analytic mechanics, are of equal dimension and dual to each other withrespect to ωM−. The point is that the ﬁrst-classness conditions in (2.6) are satisﬁed ifwe deﬁne the gauge algebra to beΓ = G≥1 + P 12 ,(3.42)by using any symplectic halving of the above kind.

It is obvious from the constructionthat the ﬁrst class constraints,J(x) = M−+ j(x)withj(x) ∈Γ⊥,(3.43)obtained by using Γ in (3.42) satisfy the suﬃcient conditions for polynomiality given inSection 3.1. With this Γ we haveΓ⊥= G≥0 + Q−12 ,(3.44a)where Q−12 is the subspace of G−12 given byQ−12 = [M−, P 12 ] .

(3.44b)By combining (3.42) and (3.44) one also easily veriﬁes the following direct sum decom-position:Γ⊥= [M−, Γ] + Ker(adM+) ,(3.45)which is just (3.14) with V = Ker(adM+). This means that (3.35) is indeed nothing butthe equation of a particular DS gauge for the ﬁrst class constraints in (3.43), as required.This special DS gauge is called the highest weight gauge [13].

Similarly as for any DSgauge, there exists therefore a basis of gauge invariant diﬀerential polynomials of thecurrent in (3.43) such that the base elements reduce to the components of jred(x) in (3.35)by the gauge ﬁxing. The KM Poisson bracket algebra of these gauge invariant diﬀerentialpolynomials is clearly identical to the Dirac bracket algebra of the corresponding current36

components, and we can thus realize the WGS -algebra as a KM Poisson bracket algebraof gauge invariant diﬀerential polynomials.The second class constraints deﬁning the highest weight gauge (3.35) are natural inthe sense that in this case τ in (3.24) runs over the basis of the space TM−= [M+ , G]which is a natural complement of KM−= Ker(adM−) in G, eq. (3.8).In the second class formalism, the conformal action generated by LM0 on the WGS -algebra is given by the following formula:δ∗f,M0 jred(x) ≡−Zdy1 f(y+) {LM0(y) , jred(x)}∗,(3.46)where the parameter function f(x+) refers to the conformal coordinate transformationδf x+ = −f(x+), cf.

(2.11), and jred(x) is to be substituted by J(x)−M−when evaluatingthe KM Poisson brackets entering into (3.46), like in (3.25). To actually evaluate (3.46),we ﬁrst replace LM0 by the objectLmod(x) = LM0(x) −12⟨M+ , J′′(x)⟩,(3.47)which is allowed under the Dirac bracket since the diﬀerence (the second term) vanishesupon imposing the constraints.

The crucial point to notice is that Lmod weakly commuteswith all the constraints deﬁning (3.35) (not only with the ﬁrst class ones) under the KMPoisson bracket.This implies that with Lmod the Dirac bracket in (3.46) is in factidentical to the original KM Poisson bracket and by this observation we easily obtainδ∗f,M0 jred(x) = f(x+) j′red(x) + f ′(x+)jred(x) + [M0, jred(x)]) −12f ′′′(x+)M+. (3.48)This proves that, with the exception of the M+-component, the sl(2) highest weightcomponents of jred(x) in (3.35) transform as conformal primary ﬁelds, whereby the con-formal content of WGS is determined by the decomposition of the adjoint of G under Sin the aforementioned manner.

We end this discussion by noting that in the highestweight gauge LM0(x) becomes a linear combination of the M+-component of jred(x) anda quadratic expression in the components corresponding to the singlets of S in G. Fromthis we see that LM0(x) and the primary ﬁelds corresponding to the sl(2) highest weightstates give a basis for the diﬀerential polynomials contained in WGS , which is thus indeeda (classical) W-algebra in the sense of the general idea in [20].In the above we proposed a ‘halving procedure’ for ﬁnding purely ﬁrst class con-straints for which WGS appears as the algebra of the corresponding gauge invariant dif-ferential polynomials. We now wish to clarify the relationship between our method and37

the construction in a recent paper by Bais et al [16], where the WGS -algebra has been de-scribed, in the special case of G = sl(n), by using a diﬀerent method. We recall that theWGS -algebra has been constructed in [16] by adding to the ﬁrst class constraints deﬁnedby the pair (G≥1, M−) the second class constraints⟨u , J(x)⟩= 0 ,for∀u ∈G 12 .

(3.49)Clearly, we recover these constraints by ﬁrst imposing our complete set of ﬁrst classconstraint belonging to (Γ, M−) with Γ in (3.42), and then partially ﬁxing the gauge byimposing the condition⟨u , J(x)⟩= 0 ,for∀u ∈Q 12 . (3.50)One of the advantages of our construction is that by using only ﬁrst class KM constraintsit is easy to construct generalized Toda theories which possess WGS as their chiral algebra,for any sl(2) subalgebra, namely by using our general method of WZNW reductions.

Thiswill be elaborated in the next chapter. We note that in [16] the authors were actuallyalso led to replacing the original constraints by a ﬁrst class system of constraints, inorder to be able to consider the BRST quantization of the theory.

For this purpose theyintroduced unphysical ‘auxiliary ﬁelds’ and thus constructed ﬁrst class constraints in anextended phase space. However, in that construction one has to check that the auxiliaryﬁelds ﬁnally disappear from the physical quantities.

Another important advantage ofour halving procedure is that it renders the use of any such auxiliary ﬁelds completelyunnecessary, since one can start by imposing a complete system of ﬁrst class constraintson the KM phase space from the very beginning. We study some aspects of the BRSTquantization in Chapter 5, and we shall see that the Virasoro central charge given in [16]agrees with the one computed by taking our ﬁrst class constraints as the starting point.The ﬁrst class constraints leading to WGS are not unique, for example one can consideran arbitrary halving in (3.41) to deﬁne Γ.

We conjecture that these W-algebras alwaysoccur under certain natural assumptions on the constraints. To be more exact, let ussuppose that we have conformally invariant ﬁrst class constraints determined by thepair (Γ, M−) where M−is a nilpotent matrix and the non-degeneracy condition (3.3)holds together with equation (3.37).

By the Jacobson-Morozov theorem, it is possibleto extend the nilpotent generator M−to an sl(2) subalgebra S = {M−, M0, M+}. It isalso worth noting that the conjugacy class of S under the automorphism group of G isuniquely determined by the conjugacy class of the nilpotent element M−.

For this andother questions concerning the theory of sl(2) embeddings into semi-simple Lie algebras38

the reader may consult refs. [32,33,38,39].

We expect that the above assumptions on(Γ, M−) are suﬃcient for the existence of a complete set of gauge invariant diﬀerentialpolynomials and their algebra is isomorphic to WGS , where M−∈S. We are not yet ableto prove this conjecture in general, but below we wish to sketch the proof in an importantspecial case which illustrates the idea.Let us assume that we have conformally invariant ﬁrst class constraints describedby (Γ, M−, H) subject to the suﬃcient conditions for polynomiality given in Section 3.1,such that H is an integral grading operator of G. We note that these are exactly theassumptions satisﬁed by the constraints in the non-degenerate case of the generalizedToda theories associated to integral gradings [18].

In this case equation (3.37) is actuallyautomatically satisﬁed as a consequence of the non-degeneracy condition (3.3). One canalso show that it is possible to ﬁnd an sl(2) algebra S = {M−, M0, M+} for which inaddition to [H, M−] = −M−one has[H, M0] = 0and[H, M+] = M+ ,(3.51)and that for this sl(2) algebra the relationKer(adM+) ⊂GH≥0(3.52)holds, where the superscript indicates that the grading is deﬁned by H. For the sl(2)subject to (3.51) the latter property is in fact equivalent to Ker(adM−) ⊂GH≤0, which isjust the non-degeneracy condition as in our case Γ = GH>0.

The proof of these statementsis given in Appendix B.We introduce a deﬁnition at this point, which will be used in the rest of the paper.Namely, we call an sl(2) subalgebra S = {M−, M0, M+} an H-compatible sl(2) fromnow on if there exists an integral grading operator H such that [H, M±] = ±M± issatisﬁed together with the non-degeneracy condition. The non-degeneracy condition canbe expressed in various equivalent forms, it can be given for example as the relation in(3.52), and its (equivalent) analogue for M−.Turning back to the problem at hand, we now point out that by using the H-compatible sl(2) we have the following direct sum decomposition of Γ⊥= GH≥0:GH≥0 = [M−, GH>0] + Ker(adM+).

(3.53)This means that the set of currents of the form (3.35) represents a DS gauge for thepresent ﬁrst class constraints.This implies the required result, that is that the W-algebra belonging to the constraints deﬁned by Γ = GH>0 together with a non-degenerate39

M−is isomorphic to WGS with M−∈S. In this example both LH(x) and LM0(x) aregauge invariant diﬀerential polynomials.

Although the spectrum of adH is integral byassumption, in some cases the H-compatible sl(2) is embedded into G in a half-integralmanner, i.e., the spectrum of adM0 can be half-integral in certain cases. We shall returnto this point later.

We further note that in general it is clearly impossible to build suchan sl(2) out of M−for which H would play the role of M0. It is possible to prove that inthose cases there is no full set of primary ﬁelds with repect to LH which would completethis Virasoro density to a generating set of the corresponding diﬀerential polynomial W-algebra.

We have seen that such a conformal basis is manifest for WGS , which seems toindicate that in the present situation the conformal structure deﬁned by the sl(2), LM0,is preferred in comparison to the one deﬁned by LH.We also would like to mention an interesting general fact about the WGS -algebras,which will be used in the next section. Let us consider the decomposition of G under thesl(2) subalgebra S. In general, we shall ﬁnd singlet states and they span a Lie subalgebrain the Lie subalgebra Ker(adM+) of G. Let us denote this zero spin subalgebra as Z. Itis easy to see that we have the semi-direct sum decompositionKer(adM+) = Z + R,[Z, R] ⊂R,[Z, Z] ⊂Z,(3.54)where R is the linear space spanned by the rest of the highest weight states, which havenon-zero spin.

It is not hard to prove that the subalgebra of the original KM algebrawhich belongs to Z, survives the reduction to WGS . In other words, the Dirac brackets ofthe Z-components of the highest weight gauge current, jred in (3.35), coincide with theiroriginal KM Poisson brackets, given by (2.1).

Furthermore, this Z KM subalgebra actson the WGS -algebra by the corresponding original KM transformations, which preservethe highest weight gauge:Jred(x) →eai(x+)ζi Jred(x) e−ai(x+)ζi + (eai(x+)ζi)′ e−ai(x+)ζi,(3.55)where the ζi form a basis of Z. In particular, one sees that the WGS -algebra inherites thesemi-direct sum structure given by (3.54) [16].

The point we wish to make is that it ispossible to further reduce the WGS -algebra by applying the general method of conformallyinvariant KM reductions to the present Z KM symmetry. In principle, one can generatea huge number of new conformally invariant systems out of the WGS -algebras in thisway, i.e., by applying conformally invariant constraints to their singlet KM subalgebras.For example, if one can ﬁnd a subalgebra of Z on which the Cartan-Killing form of Gvanishes, then one can consider the obviously conformally invariant reduction obtained40

by constraining the corresponding components of jred in (3.35) to zero. We do not explorethese ‘secondary’ reductions of the WGS -algebras in this paper.

However, their potentialimportance will be highlighted by the example of the next section.Finally, we note that, for a half-integral sl(2), one can consider (instead of usingΓ in (3.42)) also those conformally invariant ﬁrst class constraints which are deﬁned bythe triple (Γ, M0, M−) with any graded Γ for which G≥1 ⊂Γ ⊂(G≥1 + P 12 ) .Thepolynomiality conditions of Section 3.1 are clearly satisﬁed with any such non-maximalΓ, and the corresponding extended conformal algebras are in a sense between the KMand WGS -algebras.3.4. The WGS interpretation of the W ln-algebrasThe W ln-algebras are certain conformally invariant reductions of the sl(n, R) KMalgebra introduced by Bershadsky [26] using a mixed set of ﬁrst class and second classconstraints.

It is known [16] that the simplest non-trivial case W 23 , originally proposedby Polyakov [27], coincides with the WGS -algebra belonging to the highest root sl(2) ofsl(3, R). The purpose of this section is to understand whether or not these reduced KMsystems ﬁt into our framework, which is based on using purely ﬁrst class constraints, andto uncover their possible connection with the WGS -algebras in the general case.

(In thissection, G = sl(n, R).) In fact, we shall construct here purely ﬁrst class KM constraintsleading to the W ln-algebras.

The construction will demonstrate that the W ln-algebras canin general be identiﬁed as further reductions of particular WGS -algebras. The secondaryreduction process is obtained by means of the singlet KM subalgebras of the relevantWGS -algebras, in the manner mentioned in the previous section.By deﬁnition [26], the KM reduction yielding the W ln-algebra is obtained by con-straining the current to take the following form:JB(x) = M−+ jB(x),jB(x) ∈∆⊥,(3.56)where ∆denotes the set of all strictly upper triangular n × n matrices andM−= el+1,1 + el+2,2 + ... + en,n−l,(3.57)the e’s being the standard sl(n, R) generators (l ≤n −1), i.e., M−has 1’s all alongthe l-th slanted line below the diagonal.

The current in (3.56) corresponds to imposing41

the constraints φδ(x) = 0 for all δ ∈∆, like in (2.3).Generally, these constraintscomprise ﬁrst and second class parts, where the ﬁrst class part is the one belonging tothe subalgebra D of ∆deﬁned by the relation ωM−(D, ∆) = 0, (see (2.4)). The secondclass part belongs to the complementary space, C, of D in ∆.

In fact, for l = 1 theconstraints are the usual ﬁrst class ones which yield the standard W-algebras, but thesecond class part is non-empty for l > 1. The above KM reduction is so constructedthat it is conformally invariant, since the constraints weakly commute with the Virasorodensity LHl(x), see (2.10), where Hl = 1l H1 and H1 is the standard grading operator ofsl(n, R), for which [H1 , eik] = (k −i)eik.We start our construction by extending the nilpotent generator M−in (3.57) to ansl(2) subalgebra S = {M−, M0, M+}.

In fact, parametrizing n = ml + r with m = nland 0 ≤r < l, we can takeM0 = diagr timesz }| {m2 , · · ·,(l−r) timesz}|{m −12, · · ·, · · · ,r timesz}|{−m2 , · · ·,(3.58)where the mutiplicities, r and (l −r), occur alternately and end with r. The meaning ofthis formula is that the fundamental of sl(n, R) branches into l irreducible representationsunder S, r of spin m2 and l −r of spin m−12 . The explicit form of M+ is a certain linearcombination of the eik’s with (k −i) = l, which is straightforward to compute.We describe next the ﬁrst and the second class parts of the constraints in (3.56) inmore detail by using the grading deﬁned by M0.

We observe ﬁrst that in terms of thisgrading the space ∆admits the decomposition∆= ∆0 + G 12 + G1 + G>1 . (3.59)From this and the deﬁnition of ωM−, the subalgebra D comprising the ﬁrst class partcan also be decomposed intoD = D0 + D1 + G>1 ,(3.60)whereD0 = Ker (adM−) ∩∆0(3.61)is the set of the sl(2) singlets in ∆, and D1 is a subspace of G1 which we do not needto specify.

By combining (3.59) and (3.60), we see that the complementary space C, towhich the second class part belongs, has the structureC = Q0 + G 12 + P1 ,(3.62)42

where the subspace Q0 is complementary to D0 in ∆0, and P1 is complementary to D1in G1. The 2-form ωM−is non-degenerate on C by construction, and this implies by thegrading that the spaces Q0 and P1 are symplectically conjugate to each other, which isreﬂected by the notation.We shall construct a gauge algebra, Γ, so that Bershadsky’s constraints will be recov-ered by a partial gauge ﬁxing from the ﬁrst class ones belonging to Γ.

As a generalizationof the halving procedure of the previous section, we take the following ansatz:Γ = D + P 12 + P1 ,(3.63)where P 12 is deﬁned by means of some symplectic halving G 12 = P 12 + Q 12 , like in (3.41).It is important to notice that this equation can be recasted intoΓ = D0 + P 12 + G≥1 ,(3.64)which would be just the familiar formula (3.42) if D0 was not here. By using (3.57) and(3.58), D0 can be identiﬁed as the set of n×n block-diagonal matrices, σ, of the followingform:σ = block-diag{Σ0, σ0, Σ0, ....., Σ0, σ0, Σ0},(3.65)where the Σ0’s and the σ0’s are identical copies of strictly upper triangular r × r and(l −r) × (l −r) matrices respectively.

This implies thatdim D0 = 14[l(l −2) + (l −2r)2] ,(3.66)which shows that D0 is non-empty except when l = 2, r = 1, which is the case of W 2nwith n = odd. The fact that D0 is in general non-empty gives us a trouble at this stage,namely, we have now no guarantee that the above Γ is actually a subalgebra of G. Byusing the grading and the fact that D0 is a subalgebra, we see that Γ in (3.64) becomesa subalgebra if and only if[D0 , P 12 ] ⊂P 12 .

(3.67)We next show that it is possible to ﬁnd such a ‘good halving’ of G 12 for which P 12 satisﬁes(3.67).For this purpose, we use yet another grading here. This grading is provided by usingthe particular diagonal matrix, H ∈G, which we construct out of M0 in (3.58) by ﬁrstadding 12 to its half-integral eigenvalues, and then substracting a multiple of the unitmatrix so as to make the result traceless.

In the adjoint representation, we then have43

adH = adM0 on the tensors, and adH = adM0 ± 1/2 on the spinors. We notice fromthis that the H-grading is an integral grading.

In fact, the relationship between the twogradings allows us to deﬁne a good halving of G 12 as follows:P 12 ≡G 12 ∩GH1 ,andQ 12 ≡G 12 ∩GH0 . (3.68)Since M−is of grade −1 with respect to both gradings, the spaces given by (3.68) clearlyyield a sympectic halving of G 12 with respect to ωM−.

That this is a good halving, i.e.,it ensures the condition (3.67), can also be seen easily by observing that D0 has grade0 in the H-grading, too. Thus we obtain the required subalgebra Γ of G by using thisparticular P 12 in (3.64).Let us consider now the ﬁrst class constraints corresponding to the above constructedgauge algebra Γ, φγ(x) = 0 for γ ∈Γ, which bring the current into the formJΓ(x) = M−+ jΓ(x) ,jΓ(x) ∈Γ⊥.

(3.69)It is easy to verify that the original constraint surface (3.56) can be recovered from (3.69)by a partial gauge ﬁxing in such a way that the residual gauge transformations are exactlythe ones belonging to the space D. In fact, this is achieved by ﬁxing the gauge freedomcorresponding to the piece (P 12 + P1) of Γ, (3.63), by imposing the partial gauge ﬁxingconditionφqi(x) = 0 ,qi ∈(Q0 + Q 12 ),(3.70)where the qi form a basis of the space (Q0 + Q 12 ) and the φq’s are deﬁned like in (2.3).This implies that the reduced phase space deﬁned by the constraints in (3.69) is the sameas the one determined by the original constraints (3.56). In conclusion, our purely ﬁrstclass constraints, (3.69), have the same physical content as Bershadsky’s original mixedset of constraints, (3.56).Finally, we give the relationship between Bershadsky’s W ln-algebras and the sl(2)systems.

Having seen that the reduced KM phase spaces carrying the W ln-algebras canbe realized by starting from the ﬁrst class constraints in (3.69), it follows from (3.64)that the W ln-algebras coincide with particular WGS -algebras if and only if the space D0 isempty, i.e., for W 2n with n = odd. In order to establish the WGS interpretation of W ln inthe general case, we point out that the reduced phase space can be reached from (3.69)by means of the following two step process based on the sl(2) structure.

Namely, onecan proceed by ﬁrst ﬁxing the gauge freedom corresponding to the piece (P 12 +G≥1) of Γ,and then ﬁxing the rest of the gauge freedom. Clearly, the constraint surface resulting in44

the ﬁrst step is the same as the one obtained by putting to zero those components of thehighest weight gauge current representing WGS which correspond to D0. The ﬁnal reducedphase space is obtained in the second step by ﬁxing the gauge freedom generated by theconstraints belonging to D0, which we have seen to be the space of the upper triangularsinglets of S. Thus we can conclude that W ln can be regarded as a further reduction ofthe corresponding WGS , where the ‘secondary reduction’ is of the type mentioned at theend of Section 3.3.

One can exhibit primary ﬁeld bases for the W ln-algebras and describetheir structure in detail in terms of the underlying WGS -algebras by further analysing thesecondary reduction, but this is outside the scope of the present paper, see [37].45

4. Generalized Toda theoriesLet us remind ourselves that, as has been detailed in the Introduction, the standardconformal Toda ﬁeld theories can be naturally regarded as reduced WZNW theories, andas a consequence these theories possess the chiral algebras WGS × ˜WGS as their canonicalsymmetries, where S is the principal sl(2) subalgebra of the maximally non-compact realLie algebra G. It is natural to seek for WZNW reductions leading to eﬀective ﬁeld theorieswhich would realize WGS × ˜WGS as their chiral algebras for any sl(2) subalgebra S of anysimple real Lie algebra.

The main purpose of this chapter is to obtain, by combining theresults of sections 2.3 and 3.3, generalized Toda theories meeting the above requirementin the non-trivial case of the half-integral sl(2) subalgebras of the simple Lie algebras.Before turning to describing these new theories, next we brieﬂy recall the main featuresof those generalized Toda theories, associated to the integral gradings of the simple Liealgebras, which have been studied before [3,4,14-18]. The simplicity of the latter theorieswill motivate some subsequent developments.4.1.

Generalized Toda theories associated with integral gradingsThe WZNW reduction leading to the generalized Toda theories in question is set upby considering an integral grading operator H of G, and taking the special caseΓ = GH≥1and˜Γ = GH≤−1 ,(4.1)and any non-zeroM ∈GH−1and˜M ∈GH1 ,(4.2)in the general construction given in Section 2.3. We note that by an integral gradingoperator H ∈G we mean a diagonalizable element whose spectrum in the adjoint of Gconsists of integers and contains ±1, and that GHn denotes the grade n subspace deﬁnedby H. In the present case B in (2.25b) is the subalgebra GH0 of G, and, because of thegrading structure, the properties expressed by equation (2.34) hold.

Thus the eﬀectiveﬁeld equation reads as (2.37) and the corresponding action is given by the simple formulaIHeﬀ(b) = SWZ(b) −Zd2x ⟨b ˜Mb−1, M⟩,(4.3)46

where the ﬁeld b varies in the little group GH0 of H in G.Generalized, or non-Abelian, Toda theories of this type have been ﬁrst investigatedby Leznov and Saveliev [1,3], who deﬁned these theories by postulating their Lax poten-tial,AH+ = ∂+b · b−1 + M ,AH−= −b ˜Mb−1 ,(4.4)which they obtained by considering the problem that if one requires a G-valued pure-gauge Lax potential to take some special form, then the consistency of the system ofequations coming from the zero curvature condition becomes a non-trivial problem. Incomparison, we have seen in Section 2.3 that in the WZNW framework the Lax potentialoriginates from the chiral zero curvature equation (1.9), and the consistency and theintegrability of the eﬀective theory arising from the reduction is automatic.It was shown in [3,4,16] in the special case when H, M and˜M are taken to bethe standard generators of an integral sl(2) subalgebra of G, that the non-Abelian Todaequation allows for conserved chiral currents underlying its exact integrability.

Thesecurrents then generate chiral W-algebras of the type WGS , for integrally embedded sl(2)’s.By means of the argument given in Section 3.3, we can establish the structure of thechiral algebras of a wider class of non-Abelian Toda systems [18]. Namely, we see that ifM and ˜M in (4.2) satisfy the non-degeneracy conditionsKer(adM) ∩GH≥1 = {0}andKer(ad ˜M) ∩GH≤−1 = {0} ,(4.5)then the left×right chiral algebra of the corresponding generalized Toda theory is isomor-phic to WGS−× ˜WGS+, where S−(S+) is an sl(2) subalgebra of G containing the nilpotentgenerator M ( ˜M), respectively.

The H-compatible sl(2) algebras S± occurring here arenot always integrally embedded ones. Thus for certain half-integral sl(2) algebras WGScan be realized in a generalized Toda theory of the type (4.3).

As we would like to havegeneralized Toda theories which possess WGS as their symmetry algebra for an arbitrarysl(2) subalgebra, we have to ask whether the theories given above are already enoughfor this purpose or not. This leads to the technical question as to whether for everyhalf-integral sl(2) subalgebra S = {M−, M0, M+} of G there exists an integral gradingoperator H such that S is an H-compatible sl(2), in the sense introduced in Section 3.3.The answer to this question is negative, as proven in Appendix C, where the relationshipbetween integral gradings and sl(2) subalgebras is studied in detail.

Thus we have toﬁnd new integrable conformal ﬁeld theories for our purpose.47

4.2. Generalized Toda theories for half-integral sl(2) embeddingsIn the following we exhibit a generalized Toda theory possessing the left × rightchiral algebra WGS × ˜WGS for an arbitrarily chosen half-integral sl(2) subalgebra S ={M−, M0 , M+} of the arbitrary but non-compact simple real Lie algebra G. Clearly, ifone imposes ﬁrst class constraints of the type described in Section 3.3 on the currents ofthe WZNW theory then the resulting eﬀective ﬁeld theory will have the required chiralalgebra.

We shall choose the left and right gauge algebras in such a way to be dual toeach other with respect to the Cartan-Killing form.Turning to the details, ﬁrst we choose a direct sum decomposition of G 12 of the typein (3.41), and then deﬁne the induced decomposition G−12 = P−12 + Q−12 to be given bythe subspacesQ−12 ≡P⊥12 ∩G−12 = [M−, P 12 ]andP−12 ≡Q⊥12 ∩G−12 = [M−, Q 12 ] . (4.6)It is easy to see that the 2-form ωM+ vanishes on the above subspaces of G−12 as aconsequence of the vanishing of ωM−on the corresponding subspaces of G 12 .

Thus wecan take the left and right gauge algebras to beΓ = (G≥1 + P 12 )and˜Γ = (G≤−1 + P−12 ) ,(4.7)with the constant matrices M and ˜M entering the constraints given by M−and M+,respectively. The duality hypothesis of Section 2.3 is obviously satisﬁed by this construc-tion.In principle, the action and the Lax potential of the eﬀective theory can be obtainedby specializing the general formulas of Section 2.3 to the present particular case.

In ourcaseB = Q 12 + G0 + Q−12 ,(4.8)and the physical modes, which are given by the entries of b in the generalized Gaussdecomposition g = abc with a ∈eΓ and c ∈e˜Γ, are now conveniently parametrized asb(x) = exp[q 12 (x)] · g0(x) · exp[q−12 (x)] ,(4.9)where q± 12 (x) ∈Q± 12 and g0(x) ∈G0, the little group of M0 in G. Next we introducesome notation which will be useful for describing the eﬀective theory.48

The operator Adg0 maps G−12 to itself and, by writing the general element u of G−12as a two-component column vector u = (u1 u2)t with u1 ∈P−12 and u2 ∈Q−12 , we candesignate this operator as a 2 × 2 matrix:Adg0|G−12=X11(g0)X12(g0)X21(g0)X22(g0),(4.10)where, for example, X11(g0) and X12(g0) are linear operators mapping P−12 and Q−12 toP−12 , respectively. Analogously, we introduce the notationAdg−10|G 12=Y11(g0)Y12(g0)Y21(g0)Y22(g0),(4.11)which corresponds to writing the general element of G 12 as a column vector, whose upperand lower components belong to P 12 and Q 12 , respectively.The action functional of the eﬀective ﬁeld theory resulting from the WZNW reduc-tion at hand reads as follows:ISeﬀ(g0, q 12 ,q−12 ) = SWZ(g0) −Zd2x ⟨g0M+g−10, M−⟩+Zd2x⟨∂−q 12 , g0∂+q−12 g−10 ⟩+ ⟨η 12 , X−111 · η−12 ⟩,(4.12a)where the objects η± 12 ∈P± 12 are given by the formulaeη 12 = [M+, q−12 ] + Y12 · ∂−q 12andη−12 = [M−, q 12 ] −X12 · ∂+q−12 .

(4.12b)The Euler-Lagrange equation of this action is the zero curvature condition of the followingLax potential:AS+ =M−+ ∂+g0 · g−10+ g0(∂+q−12 + X−111 · η−12 )g−10,AS−= −g0M+g−10−∂−q 12 + Y −111 · η 12 . (4.13)The above new (conformally invariant) eﬀective action and Lax potential are amongthe main results of the present paper.

Clearly, for an integrally embedded sl(2) thisaction and Lax potential simplify to the ones given by equation (4.3) and (4.4).The derivation of the above formulae is not completely straightforward, and nextwe wish to sketch the main steps. First, let us remember that, by (2.29a), to specializethe general eﬀective action given by (2.40) and the Lax potential given by (2.32) to oursituation, we should express the objects ∂+cc−1 and a−1∂−a in terms of b by using the49

constraints on J and ˜J, respectively. (In the present case it would be tedious to computethe inverse matrix of Vij in (2.27), which would be needed for using directly (2.29b).

)For this purpose it turns out to be convenient to parametrize the WZNW ﬁeld g by usingthe grading deﬁned by the sl(2), i.e., asg = g+ · g0 · g−whereg+ = a · exp[q 12 ],g−= exp[q−12 ] · c . (4.14)We recall that the ﬁelds a, c, g0 and q± 12 have been introduced previously by means ofthe parametrization g = abc, with b in (4.9).

Also for later convenience, we write g± asg+ = exp[r≥1 + p 12 + q 12 ]andg−= exp[r≤−1 + p−12 + q−12 ] . (4.15)Note that here and below the subscript denotes the grade of the variables, and p± 12 ∈P± 12 .

In our case this parametrization of g is advantageous, since, as shown below, theuse of the grading structure facilitates solving the constraints.For example, the left constraint are restrictions on J<0, for which we haveJ<0 = (g+g0Ng−10 g−1+ )<0withN = ∂+g−· g−1−. (4.16)By considering this equation grade by grade, starting from the lowest grade, it is easy tosee that the constraints corresponding to G≥1 ⊂Γ are equivalent to the relationN≤−1 = g−10 M−g0 .

(4.17)The remaining left constraints set the P−12 part of J−12 to zero, and to unfold theseconstraints ﬁrst we note thatJ−12 = [p 12 + q 12 , M−] + g0 · N−12 · g−10,withN−12 = ∂+p−12 + ∂+q−12 . (4.18)By using the notation introduced in (4.10), the vanishing of the projection of J to P−12is written as[q 12 , M−] + X11 · ∂+p−12 + X12 · ∂+q−12 = 0 ,(4.19)and from this we obtain∂+p−12 = X−111 ·[M−, q 12 ] −X12 · ∂+q−12.

(4.20)Combining our previous formulae, ﬁnally we obtain that on the constraint surface of theWZNW theoryN = g−10 M−g0 + ∂+q−12 + X−111 (g0) ·[M−, q 12 ] −X12(g0) · ∂+q−12. (4.21)50

A similar analysis applied to the right constraints yields that they are equivalent to thefollowing equation:−g−1+ · ∂−g+ = −g0M+g−10−∂−q 12 + Y −111 (g0) ·[M+ , q−12 ] + Y12(g0) · ∂−q 12. (4.22)By using the relations established above, we can at this stage easily compute b−1Tb =∂+cc−1 and b ˜Tb−1 = a−1∂−a as well, and substituting these into (2.40), and using thePolyakov-Wiegmann identity to rewrite SWZ(b) for b in (4.9), results in the action in(4.12) indeed.

The Lax potential in (4.13) is obtained from the general expression in(2.32) by an additional ‘gauge transformation’ by the ﬁeld exp[−q 12 ], which made theﬁnal result simpler. Of course, for the above analysis we have to restrict ourselves to aneighbourhood of the identity where the operators X11(g0) and Y11(g0) are invertible.The choice of the constraints leading to the eﬀective theory (4.12) guarantees that thechiral algebra of this theory is the required one, WGS × ˜WGS , and thus one should be able toexpress the W-currents in terms of the local ﬁelds in the action.

To this ﬁrst we recall thatin Section 3.1 we have given an algorithm for constructing the gauge invariant diﬀerentialpolynomials W(J).The point we wish to make is that the expression of the gaugeinvariant object W(J) in terms of the local ﬁelds in (4.12) is simply W(∂+b b−1 + T(b)),where b is given by (4.9). Applying the reasoning of [40,18] to the present case, this followssince the function W is form-invariant under any gauge transformation of its argument,and the quantity (∂+b b−1 + T(b)) is obtained by a (non-chiral) gauge transformationfrom J, namely by the gauge transformation deﬁned by the ﬁeld a−1 ∈eΓ, see equations(2.31-2).

(In analogy, when considering a right moving W-current one gauge transformsthe argument ˜J by the ﬁeld c ∈e˜Γ.) We can in principle compute the object T(b), asexplained in the above, and thus we have an algorithm for ﬁnding the formulae of theW’s in terms of the local ﬁelds g0 and q± 12 .The conformal symmetry of the eﬀective theory (4.12) is determined by the left andright Virasoro densities LM0(J) and L−M0( ˜J), which survive the reduction.

To see thisconformal symmetry explicitly, it is useful to extract the Liouville ﬁeld φ by means of thedecomposition g0 = eφM0 ·ˆg0, where ˆg0 contains the generators from G0 orthogonal to M0.One can easily rewrite the action in terms of the new variables and then its conformalsymmetry becomes manifest since eφ is of conformal weight (1, 1), ˆg0 is conformal scalar,and the ﬁelds q± 12 have conformal weights ( 12, 0) and (0, 12), respectively. This assignmentof the conformal weights can be established in a number of ways, one can for examplederive it from the corresponding conformal symmetry transformation of the WZNW ﬁeldg in the gauged WZNW theory, see eq.

(5.30). We also note that the action (4.12) can be51

made generally covariant and thereby our generalized Toda theory can be re-interpretedas a theory of two-dimensional gravity since φ becomes the gravitational Liouville mode[14].We would like to point out the relationship between the generalized Toda theorygiven by (4.12) and certain non-linear integrable equations which have been associatedto the half-integral sl(2) subalgebras of the simple Lie algebras by Leznov and Saveliev,by using a diﬀerent method. (See, e.g., equation (1.24) in the review paper in J. Sov.Math.

referred to in [3].) To this we note that, in the half-integral case, one can alsoconsider that WZNW reduction which is deﬁned by imposing the left and right constraintscorresponding to the subalgebras G≥1 and G≤−1 of Γ and ˜Γ in (4.7).

In fact, the Laxpotential of the eﬀective ﬁeld theory corresponding to this WZNW reduction coincideswith the Lax potential postulated by Leznov and Saveliev to set up their theory. Thus,in a sense, their theory lies between the WZNW theory and our generalized Toda theorywhich has been obtained by imposing a larger set of ﬁrst class KM constraints.

Thismeans that the theory given by (4.12) can also be regarded as a reduction of theirtheory.There is a certain freedom in constructing a ﬁeld theory possessing the requiredchiral algebra WGS , for example, one has a freedom of choice in the halving procedureused here to set up the gauge algebra. The theories in (4.12) obtained by using diﬀerenthalvings in equation (3.41) have their chiral algebras in common, but it is not quiteobvious if these theories are always completely equivalent local Lagrangean ﬁeld theoriesor not.

We have not investigated this ‘equivalence problem’ in general.A special case of this problem arises from the fact that one can expect that in somecases the theory in (4.12) is equivalent to one of the form (4.3). This is certainly so inthose cases when for the half-integral sl(2) of M0 and M± one can ﬁnd an integral gradingoperator H such that: (i) [H , M±] = ±M±, (ii) P 12 + G≥1 = GH≥1, (iii) P−12 + G≤−1 =GH≤−1, (iv) Q−12 + G0 + Q 12 = GH0 , where one uses the M0 grading and the H-gradingon the left- and on the right hand sides of these conditions, respectively.

By deﬁnition,we call the halving G 12 = P 12 + Q 12 an H-compatible halving if these conditions aremet. (We note in passing that an sl(2) which allows for an H-compatible halving isautomatically an H-compatible sl(2) in the sense deﬁned in Section 3.3, but, as shown inAppendix C, not every H-compatible sl(2) allows for an H-compatible halving.) Thosegeneralized Toda theories in (4.12) which have been obtained by using H-compatiblehalvings in the WZNW reduction can be rewritten in the simpler form (4.3) by means52

of a renaming of the variables, since in this case the relevant ﬁrst class constraints are inthe overlap of the ones which have been considered for the integral gradings and for thehalf-integral sl(2)’s to derive the respective theories. Since the form of the action in (4.3)is much simpler than the one in (4.12), it appears important to know the list of thosesl(2) embeddings which allow for an H-compatible halving, i.e., for which conditions(i) .

. .

(iv) can be satisﬁed with some integral grading operator H and halving. We studythis group theoretic question for the sl(2) subalgebras of the maximally non-compactreal forms of the classical Lie algebras in Appendix C. We show that the existence of anH-compatible halving is a very restrictive condition on the half-integral sl(2) subalgebrasof the symplectic and orthogonal Lie algebras, where such a halving exists only for thespecial sl(2) embeddings listed at the end of Appendix C. In contrast, it turns out thatfor G = sl(n, R) an H-compatible halving can be found for every sl(2) subalgebra, sincein this case one can construct such a halving by proceeding similarly as we did in Section3.4 (see (3.68)).

This means that in the case of G = sl(n, R) any chiral algebra WGS canbe realized in a generalized Toda theory associated to an integral grading.It is interesting to observe that those theories which can be alternatively written inboth forms (4.3) and (4.12) allow for several conformal structures. This is so since in thiscase at least two diﬀerent Virasoro densities, namely LH and LM0, survive the WZNWreduction.4.3.

Two examples of generalized Toda theoriesWe wish to illustrate here the general construction of the previous section by workingout two examples. First we shall describe a generalized Toda theory associated to thehighest root sl(2) of sl(n + 2, R).

This is a half-integral sl(2) embedding, but, as weshall see explicitly, the theory (4.12) can in this case be recasted in the form (4.3), sincethe corresponding halving is H-compatible. We note that the W-algebras deﬁned bythese sl(2) embeddings have been investigated before by using auxiliary ﬁelds in [29].

Itis perhaps worth stressing that our method does not require the use of auxiliary ﬁeldswhen reducing the WZNW theory to the generalized Toda theories which possess theseW-algebras as their symmetry algebras, see also Section 5.3. According to the grouptheoretic analysis in Appendix C, the simplest case when a WGS -algebra deﬁned by ahalf-integral sl(2) embedding cannot be realized in a theory of the type (4.3) is the case53

of G = sp(4, R). As our second example, we shall elaborate on the generalized Todatheory in (4.12) which realizes the W-algebra belonging to the highest root sl(2) ofsp(4, R).i) Highest root sl(2) of sl(n + 2, R)In the usual basis where the Cartan subalgebra consists of diagonal matrices, thesl(2) subalgebra S is generated by the elementsM0 = 121· · ·000n00· · ·−1andM+ = M t−=0· · ·100n00· · ·0.

(4.23)Note that here and below dots mean 0’s in the entries of the various matrices.Theadjoint of sl(n + 2) decomposes into one triplet, 2n doublets and n2 singlets under thisS. It is convenient to parametrize the general element, g0, of the little group of M0 asg0 = eφM0 · eψT ·1.

. .00˜g000· · ·1,whereT =12 + nn· · ·00−2In00· · ·n(4.24)is trace orthogonal to M0 and ˜g0 is from sl(n).

We note that T and M0 generate thecentre of the corresponding subalgebra, G0. We consider the halving of G± 12 which isdeﬁned by the subspaces P± 12 and Q± 12 consisting of matrices of the following form:p 12 =0pt000n00· · ·0,q 12 =0· · ·000nq0· · ·0,p−12 =0· · ·0˜p0n00· · ·0,q−12 =0· · ·000n00˜q t0,(4.25)where q and ˜p are n-dimensional column vectors and pt and ˜q t are n-dimensional rowvectors, respectively.

One sees that the P and Q subspaces of G± 12 are invariant underthe adjoint action of g0, which means that the block-matrices in (4.10) and (4.11) arediagonal, and thus η± 12 = [M±, q∓12 ]. One can also verify that X11 = e−12 φ−ψ˜g0, andthat using this the eﬀective action (4.12) can be written as follows:Ieﬀ(g0, q 12 , q−12 ) = SWZ(g0) −Zd2xheφ−e−12 φ+ψ (∂+˜q)t · ˜g−10· (∂−q)+e12 φ+ψ ˜qt · ˜g−10· qi,(4.26)54

where dot means usual matrix multiplication. With respect to the conformal structuredeﬁned by M0, eφ has weights (1, 1), the ﬁelds q and ˜q have half-integer weights ( 12, 0)and (0, 12), respectively, ψ and ˜g0 are conformal scalars.

In particular, we see that φ isthe Liouville mode with respect to this conformal structure.In fact, the halving considered in (4.25) can be written like the one in (3.68), byusing the integral grading operator H given explicitly asH = M0 + 12T =1n + 2n + 100−In+1. (4.27)It is an H-compatible halving as one can verify that it satisﬁes the conditions (i) .

. .

(iv)mentioned at the end of Section 4.2, see also Appendix C. It follows that our reducedWZNW theory can also be regarded as a generalized Toda theory associated with theintegral grading H. In other words, it is possible to identify the eﬀective action (4.26)as a special case of the one in (4.3). To see this in concrete terms, it is convenient toparametrize the little group of H asb = exp(q 12 ) · g0 · exp(q−12 ),whereg0 = eΦH · eξS ·1· · ·00˜g000· · ·1,(4.28)and S = M0 −( n+22n )T is trace orthogonal to H. It is easy to check that by inserting thisdecomposition into the eﬀective action (4.3) and using the Polyakov-Wiegmann identityone recovers indeed the eﬀective action (4.26), withφ = Φ + ξandψ = 12Φ −2 + n2n ξ.

(4.29)The conformal structure deﬁned by H is diﬀerent from the one deﬁned by M0. In fact,with respect to the former conformal structure Φ is the Liouville mode and all otherﬁelds, including q and ˜q, are conformal scalars.ii) Highest root sl(2) of sp(4, R)We use the convention when the symplectic matrices have the formg =ABC−At,whereB = Bt , C = Ct ,(4.30)55

and the Cartan subalgebra is diagonal. The sl(2) subalgebra S corresponding to thehighest root of sp(4, R) is generated by the matricesM0 = 12(e11 −e33),M+ = e13 ,andM−= e31 ,(4.31)where eij denotes the elementary 4 × 4 matrix containing a single 1 in the ij-position.The adjoint of sp(4) branches into 3 + 2 · 2 + 3 · 1 under S. The three singlets generatean sl(2) subalgebra diﬀerent from S, so that the little group of M0 is GL(1) × SL(2).GL(1) is generated by M0 itself and the corresponding ﬁeld is the Liouville mode.

Usingusual Gauss-parameters for the SL(2), we can parametrize the little group of M0 asg0 = eφM010000eψ + αβe−ψ0αe−ψ00100βe−ψ0e−ψ. (4.32)We decompose the G± 12 subspaces (spanned by the two doublets) into their P and Qparts as followsp 12 + q 12 =0p0q00q0000000−p0,p−12 + q−12 =0000˜p0000˜q0−˜p˜q000.

(4.33)Now the little group, or more precisely the SL(2) generated by the three singlets, mixesthe P and Q subspaces of G−12 so that the matrices Xij and Yij in (4.10) and (4.11)possess oﬀ-diagonal elements:Xij = e−12 φeψ + αβe−ψαe−ψβe−ψe−ψ,Yij = Xji. (4.34)Inserting this into (4.12) yields the following eﬀective action:ISeﬀ(g0, q, ˜q) =SWZ(g0) −Zd2x"eφ −2e−12 φ−ψ(∂−q) · (∂+˜q)+ 2e12 φ˜q + e−12 φ−ψβ∂−q·q + e−12 φ−ψα∂+˜qeψ + αβe−ψ#,(4.35)for the Liouville mode φ, the conformal scalars ψ, α, β and the ﬁelds q, ˜q with weights( 12, 0) and (0, 12), respectively.It is easy to see directly from its formula that it is impossible to obtain the aboveaction as a special case of (4.3).

Indeed, if the expression in (4.35) was obtained from(4.3) then the non-derivative term ∼˜q q(eψ + αβe−ψ)−1 could only be gotten from thesecond term in (4.3), but, since g0 and b are matrices of unit determinant, this termcould never produce the denominator in the non-derivative term in (4.35).56

5. Quantum framework for WZNW reductionsIn this chapter we study the quantum version of the WZNW reduction by using thepath-integral formalism and also re-examine some of the classical aspects discussed inthe previous chapters.

We ﬁrst show that the conﬁguration space path-integral of theconstrained WZNW theory can be realized by the gauged WZNW theory of Section 2.2.We then point out that the eﬀective action of the reduced theory, (2.40), can be derivedby integrating out the gauge ﬁelds in a convenient gauge, the physical gauge, in whichthe gauge degrees of freedom are frozen. A nontrivial feature of the quantum theory mayappear in the path-integral measure.

We shall ﬁnd that for the generalized Toda theoriesassociated with integral gradings the eﬀective measure takes the form determined fromthe symplectic structure of the reduced theory. This means that in this case the quantumHamiltonian reduction results in the quantization of the reduced classical theory; in otherwords, the two procedures, the reduction and the quantization, commute.

We shall alsoexhibit the W-symmetry of the eﬀective action for this example. By using the gaugedWZNW theory, we can construct the BRST formalism for the WZNW reduction inthe general case.

For conformally invariant reductions, this allows for computing thecorresponding Virasoro centre explicitly. In particular, we derive here a nice formulafor the Virasoro centre of WGS for an arbitrary sl(2) embedding.

We shall verify thatour result agrees with the one obtained in [16], in spite of the apparent diﬀerence in thestructure of the constraints.5.1. Path-integral for constrained WZNW theoryIn this section we wish to set up the path-integral formalism for the constrainedWZNW theory.

For this, we recall that classically the reduced theory has been obtainedby imposing a set of ﬁrst-class constraints in the Hamiltonian formalism. Thus what weshould do is to write down the path-integral of the WZNW theory ﬁrst in phase spacewith the constraints implemented and then ﬁnd the corresponding conﬁguration spaceexpression.

The phase space path-integral can formally be deﬁned once the canonicalvariables of the theory are speciﬁed. A practical way to ﬁnd the canonical variables is thefollowing [41].

Let us start from the WZNW action SWZ(g) in (1.2) and parametrize thegroup element g ∈G in some arbitrary way, g = g(ξ). We shall regard the parameters57

ξa, a = 1, ..., dim G, as the canonical coordinates in the theory. To ﬁnd the canonicalmomenta, we introduce the 2-form A = 12Aab(ξ) dξadξb to rewrite the Wess-Zumino termas13Tr (dg g−1)3 = dA.

(5.1)The 2-form A is well-deﬁned only locally on G, since the Wess-Zumino 3-form is closedbut not exact. Fortunately we do not need to specify A explicitly below.

We next deﬁneNab(ξ) by ∂g∂ξag−1 = Nab(ξ)T b,(5.2)where T b are the generators of G. The matrix N is easily shown to be non-singular,detN ̸= 0. Upon writing SWZ(g) =Rd2x L(g), the canonical momentum conjugate to ξais found to beΠa =∂L∂∂0ξa = κhNab(ξ)(∂0g g−1)b −Aab(ξ)∂1ξbi.

(5.3)The Hamiltonian of the WZNW theory is then given by H =Rdx1H withH = Πa∂0ξa −L = 12κTrP 2 + (κ∂1g g−1)2,(5.4)whereP a = (N −1)ab(Πb + κAbc∂1ξc). (5.5)Since P = κ∂0g g−1 in the original variables, the Hamiltonian density takes the usualSugawara form as expected.Classically, the constrained WZNW theory has been deﬁned as the usual WZNWtheory with its KM phase space reduced by the set of constraints given by (2.16), whichin the canonical variables readφi = ⟨γi, P + κ(∂1g g−1 −M)⟩= 0,˜φi = ⟨˜γi, g−1Pg −κ(g−1∂1g + ˜M)⟩= 0,(5.6)with the bases γi ∈Γ, ˜γi ∈˜Γ.

As in Section 2.2, no relationship is assumed here betweenthe two subalgebras, Γ and ˜Γ. Now we write down the phase space path-integral for theconstrained WZNW theory.

According to Faddeev’s prescription [42] it is deﬁned asZ =ZdΠdξ δ(φ)δ(˜φ)δ(χ)δ(˜χ) det |{φ, χ}| det |{˜φ, ˜χ}|× expiZd2x (Πa∂0ξa −H),(5.7)58

where we implement the ﬁrst class constraints by inserting δ(φ) and δ(˜φ) in the path-integral. The δ-functions of χ and ˜χ refer to gauge ﬁxing conditions corresponding tothe constraints, φ and ˜φ, which act as generators of gauge symmetries.

By introducingLagrange-multiplier ﬁelds, A−= Ai−γi and A+ = Ai+˜γi, (5.7) can be written asZ =ZdΠdξdA+dA−δ(χ)δ(˜χ) det |{φ, χ}| det |{˜φ, ˜χ}|× expiZd2xTr (Π∂0ξ + A−φ + A+ ˜φ) −H. (5.8)By changing the momentum variable from Πa to P a in (5.5), the measure acquires adeterminant factor, dΠ = dP det N, and the integrand of the exponent in (5.8) becomesTr (Π∂0ξ + A−φ + A+ ˜φ) −H= κTrh−12 1κP2 + 1κP(A−+ gA+g−1 + ∂0g g−1) −N −1A ∂1ξ(∂0g g−1)−12(∂1g g−1)2 + A−(∂1g g−1 −M) −A+(g−1∂1g + ˜M)i.

(5.9)Since the matrix N(ξ) is independent of P, we can easily perform the integration overP provided that the remaining δ-functions and the determinant factors are also P-independent. We can choose the gauge ﬁxing conditions, χ and ˜χ, so that this is true.

(For example, the physical gauge which we will choose in the next section fulﬁlls thisdemand.) Then we end up with the following formula of the conﬁguration space path-integral:Z =Zdξ det N dA+dA−δ(χ)δ(˜χ) det |{φ, χ}| det |{˜φ, ˜χ}| eiI(g,A−,A+),(5.10)where I(g, A−, A+) is the gauged WZNW action (2.18).

We note that the measure forthe coordinates in this path-integral is the invariant Haar measure,dµ(g) =Yadξa det N =Ya(dg g−1)a. (5.11)This is a consequence of the fact that the phase space measure in (5.7) is invariant undercanonical transformations to which the group transformations belong.The above formula for the conﬁguration space path-integral means that the gaugedWZNW theory provides the Lagrangian realization of the Hamiltonian reduction, whichwe have already seen on the basis of a classical argument in Section 2.2.59

5.2. Eﬀective theory in the physical gaugeHaving seen how the constrained WZNW theory is realized as the gauged WZNWtheory, we next discuss the eﬀective theory which arises when we eliminate all the un-physical degrees of freedom in a particularly convenient gauge, the physical gauge.

Weshall rederive, in the path-integral formalism, the eﬀective action which appeared in theclassical context earlier in this paper. For this purpose, within this section we restrictour attention to the left-right dual reductions considered in Section 2.3.

It, however,should be noted that this restriction is not absolutely necessary to get an eﬀective ac-tion by the method given below. In this respect, it is also worth noting that Polyakov’s2-dimensional gravity action in the light-cone gauge can be regarded as an eﬀective ac-tion in a non-dual reduction, which is obtained by imposing a constraint only on theleft-current for G = SL(2) [43,12].

We will not pursue the non-dual cases here.To eliminate all the unphysical gauge degrees of freedom, we simply gauge themaway from g, i.e., we gauge ﬁx the Gauss decomposed g in (2.25) into the formg = abc →b. (5.12)More speciﬁcally, with the parametrization a(x) = exp [σi(x)γi], c(x) = exp [˜σi(x)˜γi] wedeﬁne the physical gauge byχi = σi = 0,˜χi = ˜σi = 0.

(5.13)We here note that for this gauge the determinant factors in (5.8) are actually constants.Now the eﬀective action is obtained by performing the A± integrations in (5.10). Theintegration of A−gives rise to the delta-function,Yiδ⟨γi, bA+b−1 + ∂+b b−1 −M⟩,(5.14)with γi ∈Γ normalized by the duality condition (2.22).

One then notices that the delta-function (5.14) implies exactly condition (2.29) with ∂+c c−1 replaced by A+. Hence,with the help of the matrix Vij(b) in (2.27) and T(b) in (2.29), it can be rewritten as(det V )−1 δA+ −b−1T(b)b.

(5.15)Finally, the integration of A+ yieldsZ =Zdµeﬀ(b) eIeff(b),(5.16)60

where Ieﬀ(b) is the eﬀective action (2.40)*, and dµeﬀ(b) is the eﬀective measure given bydµeﬀ(b) = (det V )−1 dµ(g)δ(σ)δ(˜σ) = (det V )−1 dµ(g)dσd˜σσ=˜σ=0. (5.17)Of course, as far as the eﬀective action is concerned, the path-integral approachshould give the same result as the classical one, because the integration of the gaugeﬁelds is Gaussian and hence equivalent to the classical elimination of the gauge ﬁeldsby their ﬁeld equations.

However, a non-trivial feature may arise at the quantum levelwhen the eﬀective path-integral measure (5.17) is taken into account. Let us examine theeﬀective measure in the simple case where the space B = (Γ + ˜Γ)⊥, with which b ∈eB,forms a subalgebra of G satisfying (2.34), and thus the eﬀective action in (5.16) simpliﬁestoIeﬀ(b) = SWZ(b) −κZd2x ⟨b ˜Mb−1, M⟩.

(5.18)In this case, the 1-form appearing in the measure dµ(g) of (5.11),dg g−1 = da a−1 + a(db b−1)a−1 + ab(dc c−1)b−1a−1,(5.19)turns out, in the physical gauge, to bedg g−1σ=˜σ=0 = γidσi + db b−1 + Vij(b)˜γid˜σj. (5.20)As a result, the determinant factor in (5.17) is cancelled by the one coming from (5.20),and the eﬀective measure admits a simple form:dµeﬀ(b) = db b−1.

(5.21)The point is that this is exactly the measure which is determined from the symplecticstructure of the eﬀective theory (5.18) obtained by the classical Hamiltonian reduction.This tells us that in this case the quantum Hamiltonian reduction results in the quanti-zation of the reduced classical theory. In particular, since the above assumption for B issatisﬁed for the generalized Toda theories associated with integral gradings, we concludethat these generalized Toda theories are equivalent to the corresponding constrained* Actually, the eﬀective action always takes the form (2.40) if one restricts the WZNWﬁeld to be of the form g = abc with a ∈eΓ, c ∈e˜Γ and b such that Vij(b) is invertible.The duality between Γ and ˜Γ is not necessary but can be used to ensure this technicalassumption.61

(gauged) WZNW theories even at the quantum level, i.e., including the measure. Thisresult has been established before in the special case of the standard Toda theory (1.1)in [44], where the measure dµeﬀ(b) is simply given by Qi dϕi.We end this section by noting that it is not clear whether the measure determinedfrom the symplectic structure of the reduced classical theory is identical to the eﬀectivemeasure (5.17) in general.

In the general case both measures in question could becomequite involved and thus one would need some geometric argument to see if they areidentical or not.5.3. The W-symmetry of the generalized Toda action IHeﬀ(b)In the previous section we have seen the quantum equivalence of the generalizedToda theories given by (4.3) and the corresponding constrained WZNW theories.Itfollows from their WZNW origin that the generalized Toda theories possess conservedW-currents.

It is thus natural to expect that their eﬀective actions, IHeﬀin (4.3) and ISeﬀin(4.12), allow for symmetry transformations yielding the W-currents as the correspondingNoether currents. We demonstrate below that this is indeed the case on the example ofthe theories associated with integral gradings, when the action takes a simple form.

Wehowever believe that there are symmetries of the eﬀective action corresponding to theconserved chiral currents inherited from the KM algebra for any reduced WZNW theory.Let us consider a gauge invariant diﬀerential polynomial W(J) in the constrainedWZNW theory giving rise to the eﬀective theory described by the action in (4.3). In termsof the generalized Toda ﬁeld b(x), this conserved W-current is given by the diﬀerentialpolynomialWeﬀ(β) = W(M + β),whereβ ≡∂+b b−1.

(5.22)This equality [34,15] holds because the constrained current J and (M + β) (which is,incidentally, just the Lax potential AH+ in (4.4)) are related by a gauge transformation,as we have seen. By choosing some test function f(x+), we now associate to Weﬀ(β) thefollowing transformation of the ﬁeld b(x):δW b(y) =hZd2x f(x+)δWeﬀ(x)δβ(y)i· b(y) ,(5.23)and we wish to show that δW b is a symmetry of the action IHeﬀ(b).

Before proving this, we62

notice, by combining the deﬁnition in (5.23) with (5.22), that (δW b)b−1 is a polynomialexpression in f, β and their ∂+-derivatives up to some ﬁnite order.We start the proof by noting that the change of the action under an arbitraryvariation δb is given by the formulaδIHeﬀ(b) = −Zd2y ⟨δb b−1(y) , b(y) δIHeﬀδb(y)⟩= −Zd2y ⟨δb b−1(y) , ∂−β(y) + [b(y) ˜Mb−1(y), M]⟩. (5.24)In the next step, we use the ﬁeld equation to replace ∂−β by −[b ˜Mb−1, M] in the obviousequality∂−Weﬀ(x) =Zd2y ⟨δWeﬀ(x)δβ(y) , ∂−β(y)⟩,(5.25)and then, from the fact that ∂−Weﬀ= 0 on-shell, we obtain the following identity:Zd2y ⟨δWeﬀ(x)δβ(y) , [b(y) ˜Mb−1(y), M]⟩= 0 ,(5.26)Of course, the previous argument only implies that (5.26) holds on-shell.

However, wenow make the crucial observation that (5.26) is an oﬀ-shell identity, i.e., it is valid for anyﬁeld b(x) not only for the solutions of the ﬁeld equation. This follows by noticing thatthe object in (5.26) is a local expression in b(x) containing only x+-derivatives.

In fact,any such object which vanishes on-shell has to vanish also oﬀ-shell, because one can ﬁndsolutions of the ﬁeld equation for which the x+-dependence of the ﬁeld b is prescribed inan arbitrary way at an arbitrarily chosen ﬁxed value of x−.By using the above observation, it is easy to show that δW b in (5.23) is indeed asymmetry of the action. First, simply inserting (5.23) into (5.24), we haveδW IHeﬀ(b) = −Zd2x f(x+)Zd2y ⟨δWeﬀ(x)δβ(y), ∂−β(y) + [b(y) ˜Mb−1(y), M]⟩.

(5.27)We then rewrite this equation asδW IHeﬀ(b) = −Zd2x f(x+)∂−Weﬀ(x),(5.28)with the aid of the identities (5.26) and (5.25). This then proves thatδW IHeﬀ(b) = 0 ,(5.29)63

since the integrand in (5.28) is a total derivative, thanks to ∂−f = 0. One can alsosee, from equation (5.23), that Weﬀis the Noether charge density corresponding to thesymmetry transformation δW b of IHeﬀ(b).5.4.

BRST formalism for WZNW reductionsSince the constrained WZNW theory can be regarded as the gauged WZNW theory(2.18), one is naturally led to construct the BRST formalism for the theory as a basis forquantization. Below we discuss the BRST formalism based on the gauge symmetry (2.19)and thus return to the general situation of Section 5.1 where no relationship between thetwo subalgebras, Γ and ˜Γ, is supposed.Prior to the construction we here note how the conformal symmetry is realized inthe gauged WZNW theory when there is an operator H satisfying the condition (2.13).

(For simplicity, in what follows we discuss the symmetry associated to the left-movingsector. )In fact, with such H and a chiral test function f +(x+) one can deﬁne thefollowing transformation,δg = f +∂+g + ∂+f +Hg,δA−= f +∂+A−+ ∂+f +[H, A−],δA+ = f +∂+A+ + ∂+f +A+,(5.30)which leaves the gauged WZNW action I(g, A−, A+) invariant.

This corresponds exactlyto the conformal transformation in the constrained WZNW theory generated by theVirasoro density LH in (2.10), as can be conﬁrmed by observing that (5.30) implies theconformal action (2.11) for the current with f(x+) = f +(x+). We shall derive later theVirasoro density as the Noether charge density in the BRST system.Turning to the construction of the BRST formalism, we ﬁrst choose the space Γ∗⊂Gwhich is dual to Γ with respect to the Cartan-Killing form (and similarly ˜Γ∗dual to ˜Γ).Following the standard procedure [45] we introduce two sets of ghost, anti-ghost andNakanishi-Lautrup ﬁelds, {c ∈Γ, ¯c+, B+ ∈Γ∗} and {b ∈˜Γ, ¯b−, B−∈˜Γ∗}.

The BRSTtransformation corresponding to the (left-sector of the) local gauge transformation (2.19)64

is given byδBg = −cg,δBA−= D−c,δBc = −c2,δB¯c+ = iB+,δBB+ = 0,δB(others) = 0,(5.31)with D± = ∂±∓[A±,]. After deﬁning the BRST transformation ¯δB for the right-sectorin an analogous way, we write the BRST action by adding a gauge ﬁxing term and aghost term to the gauged action,IBRST = I(g, A−, A+) + Igf + Ighost.

(5.32)The additional terms can be constructed by the manifestly BRST invariant expression,Igf+Ighost = −iκ(δB + ¯δB)Zd2x⟨¯c+, A−⟩+ ⟨¯b−, A+⟩= κZd2x⟨B+, A−⟩+ ⟨B−, A+⟩+ i⟨¯c+, D−c⟩+ i⟨¯b−, D+b⟩,(5.33)where we have chosen the gauge ﬁxing conditions as A± = 0. Then the path-integral forthe BRST system is given byZ =Zdµ(g) dA+dA−dc d¯c+db d¯b−dB+dB−eiIBRST,(5.34)which, upon integration of the ghosts and the Nakanishi-Lautrup ﬁelds, reduces to (5.10).

(Strictly speaking, for this we have to generalize the gauge ﬁxing conditions in (5.10) tobe dependent on the gauge ﬁelds.) By this construction the nilpotency, δ2B = 0, and theBRST invariance of the action, δBIBRST = 0, are easily checked.It is, however, convenient to deal with the simpliﬁed BRST theory obtained byperforming the trivial integrations of A± and B± in (5.34),IBRST(g, c, ¯c+, b,¯b−) = SWZ(g) + iκZd2x⟨¯c+, ∂−c⟩+ ⟨¯b−, ∂+b⟩.

(5.35)We note that this eﬀective BRST theory is not merely a sum of a free WZNW sector andfree ghost sector as it appears, but rather it consists of the two interrelated sectors inthe physical space speciﬁed by the BRST charge deﬁned below. At this stage the BRSTtransformation which leaves the simpliﬁed BRST action (5.35) invariant readsδBg = −cg,δBc = −c2,δB¯c+ = −πΓ∗hi(∂+g g−1 −M−) + (c¯c+ + ¯c+c)i,δB(others) = 0,(5.36)65

where πΓ∗= Pi |γ∗i ⟩⟨γi| is the projection operator onto the dual space Γ∗with thenormalized bases, ⟨γi, γ∗j ⟩= δij. From the associated conserved Noether current, ∂−jB+ =0, the BRST charge QB is deﬁned to beQB =Zdx+jB+(x) =Zdx+⟨c, ∂+g g−1 −M −c¯c+⟩.

(5.37)The physical space is then speciﬁed by the condition,QB|phys⟩= 0. (5.38)In the simple case of the WZNW reduction which leads to the standard Toda theory, theBRST charge (5.37) agrees with the one discussed earlier [46].In the case where there is an H operator which guarantees the conformal invariance,the BRST system also has the corresponding conformal symmetry,δg = f +∂+g + ∂+f +Hg,δc = f +∂+c + ∂+f +[H, c],δ¯c+ = f +∂+¯c+ + ∂+f +(¯c+ + [H, ¯c+]),δb = f +∂+b,δ¯b−= f +∂+¯b−,(5.39)inherited from the one (5.30) in the gauged WZNW theory.

If the H operator furtherprovides a grading, one ﬁnds from (5.39) that the currents of grade −h have the (left-)conformal weight 1−h, except the H-component, which is not a primary ﬁeld. Similarly,the ghosts c, ¯c+ of grade h, −h have the conformal weight h, 1 −h, respectively, whereasthe ghosts b, ¯b are conformal scalars.

Now we deﬁne the total Virasoro density operatorLtot from the associated Noether current, ∂−jC+ = 0, byZdx+jC+(x) = 1κZdx+f +(x+)Ltot(x). (5.40)The (on-shell) expression is found to be the sum of the two parts, Ltot = LH + Lghost,where LH is indeed the Virasoro operator (2.10) for the WZNW part, andLghost = iκ⟨¯c+, ∂+c⟩+ ∂+⟨H, c¯c+ + ¯c+c⟩,(5.41)is the part for the ghosts.

The conformal invariance of the BRST charge, δQB = 0, orequivalently, the BRST invariance of the total conformal charge, δBLtot = 0, are readilyconﬁrmed.66

Let us ﬁnd the Virasoro centre of our BRST system. The total Virasoro centre ctotis given by the sum of the two contributions, c from the WZNW part and cghost from theghost one.

The Viraso centre from LH is given byc = k dim Gk + g−12k⟨H, H⟩,(5.42)where k is the level of the KM algebra and g is the dual Coxeter number. On the otherhand, the ghosts contribute to the Virasoro centre by the usual formula,cghost = −2XΓ1 + 6h(h −1),(5.43)where the summation is performed over the eigenvectors of adH in the subalgebra Γ.

(One can conﬁrm (5.43) by performing the operator product expansion with Lghost in(5.41).)5.5. The Virasoro centre in two examplesBy elaborating on the general result of the previous section, we here derive explicitformulas for the total Virasoro centre in two important special cases of the WZNWreduction.i) The generalized Toda theory IHeﬀ(b)In this case the summation in (5.43) is over the eigenstates of adH with eigenvaluesh > 0, since Γ = GH>0.

We can establish a concise formula for ctot, (5.46) below, by usingthe following group theoretic facts.First, we can assume that the grading operator H ∈G is from the Cartan subalgebraof the complex simple Lie algebra Gc containing G. Second, the scalar product ⟨, ⟩deﬁnes a natural isomorphism between the Cartan subalgebra and the space of roots,and we introduce the notation ⃗δ for the vector in root space corresponding to H underthis isomorphism.More concretely, this means that we set H = Pi δiHi by usingan orthonormal Cartan basis, ⟨Hi, Hj⟩= δij. Third, we recall the strange formula ofFreudenthal-de Vries [47], which (by taking into account the normalization of ⟨, ⟩andthe duality between the root space and the Cartan subalgebra) readsdim G = 12g |⃗ρ|2 ,(5.44)67

where ⃗ρ is the Weyl vector, given by half the sum of the positive roots. Fourth, we choosethe simple positive roots in such a way that the corresponding step operators, which arein general in Gc and not in G, have non-negative grades with respect to H.By using the above conventions, it is straightforward to obtain the following expres-sionsXh>01 = dim Γ = 12(dim G −dim GH0 ),Xh>0h = 2(⃗ρ · ⃗δ),Xh>0h2 = 12tr (adH)2 = g⟨H, H⟩= g|⃗δ|2,(5.45)for the corresponding terms in (5.43).

Substituting these into (5.43) and also (5.44) into(5.42),one can ﬁnally establish the following nice formula of the total Virasoro centre [14]:ctot = c + cghost = dim GH0 −12pk + g ⃗δ −1√k + g ⃗ρ2. (5.46)In particular, in the case of the reduction leading to the standard Toda theory (1.1) theresult (5.46) is consistent with the one directly obtained in the reduced theory [8,10].ii) The WGS -algebra for half-integral sl(2) embeddingsFor sl(2) embeddings the role of the H is played by M0 and in the half-integral casewe have Γ = G≥1 + P 12 = G>0 −Q 12 .

It follows that the value of the total Virasorocentre can now be obtained by substracting the contribution of the ‘missing ghosts’corresponding to Q 12 , which is 12dim G 12 , from the expression in (5.46). We thus obtainthat in this casectot = Nt −12Ns −12pk + g ⃗δ −1√k + g ⃗ρ2,(5.47a)whereNt = dim G0 ,andNs = dim G 12 ,(5.47b)are the number of tensor and spinor multiplets in the decomposition of the adjoint of Gunder the sl(2) subalgebra S, respectively.

We note that, as proven by Dynkin [39], it ispossible to choose a system of positive simple roots so that the grade of the correspondingstep operators is from the set {0, 12, 1}, and that ⃗δ is ( 12×) the so called deﬁning vectorof the sl(2) embedding in Dynkin’s terminology.As has been mentioned in Section 3.3, Bais et al [16] (see also [29]) studied a similarreduction of the KM algebra for half-integral sl(2) embeddings where all the current68

components corresponding to G>0 are constrained from the very beginning.In theirsystem, the constraints (3.49) of G 12 , being inevitably second-class, are modiﬁed intoﬁrst-class by introducing an auxiliary ﬁeld to each constraint of G 12 . Accordingly, theauxiliary ﬁelds give rise to the extra contribution −12dim G 12 in the total Virasoro centre.It is clear that adding this to the sum of the WZNW and ghost parts (which is of theform (5.46) with M0 substituted for H), renders the total Virasoro centre of their systemidentical to that of our system, given by (5.47).

This result is natural if we recall the factthat their reduced phase space (after complete gauge ﬁxing) is actually identical to ours.It is obvious that our method, which is based on purely ﬁrst-class KM constraints anddoes not require auxiliary ﬁelds, provides a simpler way to reach the identical reducedtheory.69

6. DiscussionThe main purpose of this paper has been to study the general structure of theHamiltonian reductions of the WZNW theory.

Considering the number of interestingexamples resulting from the reduction, this problem appears important for the theory oftwo-dimensional integrable systems and in particular for conformal ﬁeld theory.Our most important result perhaps is that we established the gauged WZNW settingof the Hamiltonian reduction by ﬁrst class constraints in full generality. It was then usedhere to set up the BRST formalism in the general case, and for obtaining the eﬀectiveactions for the left-right dual reductions.

We hope that the general framework we set upwill be useful for further studies of this very rich problem.The other major concern of the paper has been to investigate the W-algebras andtheir ﬁeld theoretic realizations arising from the WZNW reduction. We found ﬁrst classKM constraints leading to the WGS -algebras which allowed us to construct generalizedToda theories realizing these interesting extended conformal algebras.

We believe thatthe sl(2)-embeddings underlying the WGS -algebras are to play an important organizingrole in general for understanding the structure, especially the primary ﬁeld content, ofthe conformally invariant reduced KM systems.We illustrated this idea by showingthat the W ln-algebras are nothing but further reductions of WGS -algebras belonging toparticular sl(2)-embeddings (see also [37]). In our study of W-algebras we employedtwo (apparently) new methods, which are likely to have a wider range of applicabilitythan what we exploited here.

The ﬁrst is the method of symplectic halving whereby weconstructed purely ﬁrst class KM constraint for the WGS as well as for the W ln-algebras.The second is what we call the sl(2)-method, which can be summarized by saying thatif one has conformally invariant ﬁrst class constraints given by some (Γ, M−) with M−nilpotent, then one should build the sl(2) containing M−and try to analyse the systemin terms of this sl(2). We used this method to investigate, in the non-degenerate case,the generalized Toda sytems belonging to integral gradings, and also to provide theWGS -interpretation of the W ln-algebras.We wish to remark here that, as far as we know, the technical problem concerningthe inequivalence of those WGS -algebras which belong to group theoretically inequivalentsl(2) embeddings has not been tackled yet.It is well known [22] that the standard W-algebras can be identiﬁed as the second70

Poisson bracket structure of the generalized KdV hierarchies of Drinfeld-Sokolov [5]. Asimilar relationship between W-algebras and KdV type hierarchies has been establishedvery recently in more general cases [28,48,49].

In particular, the W ln-algebras have beenrelated to the so called fractional KdV hierarchies. It would be clearly worthwhile tostudy in general the relationship between the generalized Drinfeld-Sokolov hierachies of[48] and the WGS -algebras together with their further reductions, see also [16,17].We gave a general local analysis of the eﬀective theories arising in the left-rightdual case of the reduction, and investigated in particular the generalized Toda theoriesobtained by the reduction in some detail.

In the case of the generalized Toda theoriesassociated with the integral gradings we exhibited the way in which the W-symmetryoperates as an ordinary symmety of the action, and demonstrated that the quantumHamiltonian reduction is consistent with the canonical quantization of the reduced clas-sical theory. It would be nice to have the analogous problems under control also in moregeneral cases.

In our analysis we restricted the considerations to Gauss-decomposableﬁelds. The fact that the Gauss decomposition may break down can introduce apparentsingularities in the local description of the eﬀective theories, but the WZNW descriptionis inherently global and remains valid for non Gauss-decomposable ﬁelds as well [12,13].It is hence an interesting problem to further analyze the global (topological) aspects ofthe phase space of the reduced WZNW theories.We should also note that it is possible to remove the technical assumption of left-right duality.

In particular, the study of purely chiral WZNW reductions could be ofimportance, as they are likely to give natural generalizations of Polyakov’s 2d gravityaction [43,12].In this paper we assumed the existence of a gauge invariant Virasoro density LH, ofthe form given by (2.10), for obtaining conformally invariant reductions. Based on thisassumption, we came to realize that, when H provides a grading of Γ and M, the sl(2)built out of M = M−plays an important role.

However, the example of Appendix Aindicates that there is another class of conformally invariant reductions where the formof the surviving Virasoro density is diﬀerent from that of an LH. The study of this novelway of preserving the conformal invariance may open up a new perspective on conformalreductions of the WZNW theory as well as on W-algebras.There are many further interesting questions related to the Hamiltonian reductionsof the WZNW theory, which we could not mention in this paper.

We hope to be able topresent those in future publications.71

Acknowledgement.We wish to thank B. Spence for a suggestion which has beencrucial for us for understanding the W-symmetry of the Toda action.Note added. After ﬁnishing this paper, there appeared a preprint [50] also advocatingthe importance of sl(2) structures in classifying W-algebras.72

Appendix A: A solvable but not nilpotent gauge algebraIn all the cases of the reduction we considered in Chapters 3 and 4, the gauge algebraΓ was a graded nilpotent subalgebra of G. On the other hand, we have seen in Section2.1 that the ﬁrst-classness of the constraints imply that Γ is solvable. We want hereto discuss a constrained WZNW model for which the gauge algebra is solvable but notnilpotent.

Interestingly enough, it turns out that in this example no H satisfying (2.13)exists which would render the constraints conformally invariant. However, conformalinvariance can still be maintained, showing clearly that the existence of such an H isonly a suﬃcient but not a necessary condition.We choose the Lie algebra G to be sl(3, R) and the gauge algebra Γ as generated bythe following three generatorsγ1 = Eα1 =010000000,γ2 = Eα1+α2 =001000000,(A.1a)γ3 =1√3(2H1 + H2) + 12(Eα2 −E−α2) =1√3000−12√3120−12−12√3,(A.1b)where the Cartan-Weyl generators are normalized by [Hi, E±αi]=±E±αi and[Eαi, E−αi] = 2Hi, for the simple positive roots αi.Note that, being diagonalizableover the complex numbers, γ3 is not a nilpotent operator.

The algebra of Γ is[γ1, γ2] = 0,[γ1, γ3] = −√32 γ1 + 12γ2,[γ2, γ3] = −12γ1 −√32 γ2. (A.2)It is easy to verify that Γ is a solvable, not-nilpotent Lie algebra.

It qualiﬁes as a gaugealgebra since Tr (γi γj) = 0.It is readily checked that the spaces Γ⊥and [Γ, Γ]⊥are given byΓ⊥= span{H2, Eα1, Eα1+α2, 2H1 +√3Eα2, 2H1 −√3E−α2},[Γ, Γ]⊥= span{H1, H2, Eα1, Eα1+α2, Eα2, E−α2}. (A.3)Thus [Γ, Γ]⊥/Γ⊥, which is the space of the M’s leading to ﬁrst class constraints, isone-dimensional, and we can takeM = µY ≡µ√3(4H1 + 2H2) = µ√32000−1000−1(A.4)73

without loss of generality.The next question is the conformal invariance. As discussed in Section 2.1, a suf-ﬁcient condition for conformal invariance is provided by the existence of a (modiﬁed)Virasoro density LH = LKM −∂x⟨H, J(x)⟩weakly commuting with the constraints.

Forthis to work, the generator H must satisfy the three conditions in (2.13). However, it isan easy matter to show that those conditions are contradictory in the present case, andtherefore no such H exists.The above analysis can also be carried out for the simpler gauge algebra spannedby γ3 only.

This gauge algebra is obviously nilpotent, since it is Abelian. Nevertheless,the previous conclusions remain: There exists no H which would render the ﬁrst classconstraints conformally invariant, for any M ̸= 0 from [Γ, Γ]⊥/Γ⊥.This shows theimportance of the gauge generators being nilpotent operators, rather than the gaugealgebra being nilpotent.

It would be interesting to know whether there is always an Hsatisfying (2.13) for gauge algebras consisting of nilpotent operators.Although there is no H such that the constraints are preserved by LH, we cannevertheless construct another Virasoro density Λ which does preserve the constraints.It is given byΛ(x) = LKM(x) −µ⟨γt3, J(x)⟩. (A.5)For M given in (A.4), the constraints read⟨γ1, J(x)⟩= ⟨γ2, J(x)⟩= 0 ,⟨γ3, J(x)⟩= µ ,(A.6)and are checked to weakly commute with Λ: {Λ(x), ⟨γi, J(y)⟩} ≈0 on the constraintsurface (A.6).

(Note that, when going from LKM to Λ, we have not changed the conformalcentral charge, which is classically zero.) Therefore we expect the reduced theory to beinvariant under the conformal transformation generated by Λ being its Noether chargedensity.

We now proceed to show that it is indeed the case. Before doing this, we displaythe form of Λ on the constraint surface:Λ(x) = T 21 (x) + T 22 (x) ,(A.7a)T1 = 12⟨Eα2 + E−α2, J⟩,T2 = ⟨H2, J⟩.

(A.7b)Following the analysis of Section 2.3, we take the left and right gauge algebras to bedual to each other (⟨γi, ˜γj⟩= δij)Γ = span{γ1, γ2, γ3},˜Γ = span{˜γ1, ˜γ2, ˜γ3} = span{γt1, γt2, γt3},(A.8)74

and consider M = µY and ˜M = νY t = νY . We write the SL(3, R) group elements asg = a · b · c, with a ∈exp Γ, b ∈exp H and c ∈exp ˜Γ, with H = span{Y, H2} the Cartansubalgebra.

We did not conform to the general prescription given in Section 2.3, whichrequired to write g = abc with b ∈exp B for a space B complementary to Γ + ˜Γ in G,eqs.(2.25-26). Had we done that, the resulting eﬀective action would have looked muchmore complicated.

Here, we simply take a set of coordinates in which the action lookssimple.The reduction yields an eﬀective theory for the group-valued ﬁeld b, of which theeﬀective action is given by (2.40) with (2.29b). Using the parametrization b = exp (αY )·exp (2βH2), the explicit form of the eﬀective action isIeﬀ(α, β) =Zd2xn∂+α∂−α + ∂+β∂−β −(∂+α −µ)(∂−α −ν)cosh2 βo.

(A.9)By inspection, we see that this eﬀective action is going to be conformally invariant if theﬁeld β is a scalar, and if the transformation of α is such that µ −∂+α and ν −∂−α are(1,0) and (0,1) vectors respectively. It implies that, under a conformal transformationx± −→x± −f ±(x±), the ﬁelds α and β transform asδα = f + (∂+α −µ) + f −(∂−α −ν),δβ = f + ∂+β + f −∂−β.

(A.10)We now want to show our previous claim: the action (A.9) is conformally invari-ant under the conserved Virasoro density Λ(x), which reproduces the f +-transformations(A.10) by Poisson brackets. (The f −-transformations could also be realized by construct-ing the corresponding Virasoro density ˜Λ in the right-handed sector in a similar way.

)For this, we ﬁrst note that in terms of the reduced variables α and β the two currentcomponents T1 and T2 of (A.7b) readT1 = −(µ −∂+α) tanh β ,andT2 = ∂+β. (A.11)These expressions can be obtained as follows.

Writing g = a·b·c and using the constraints(2.29b), the constrained current readsJ = a[T(b) + ∂+b · b−1]a−1 + ∂+a · a−1,(A.12)with T(b) given by (2.29). Although neither T1 nor T2 is gauge invariant, the quantitywe want to compute, Λ(x), is gauge invariant.

As a result, it cannot depend on the gauge75

variables contained in a. Hence we can just as well put a = 1 in (A.12).

Doing that, thedeﬁnitions (A.7b) yield (A.11). We thus ﬁnd the following expression for Λ:Λ = (µ −∂+α)2 tanh2 β + (∂+β)2.

(A.13)It is an easy matter to show, by using the ﬁeld equations obtained from the action (A.9),sinh2 β ∂+∂−α + tanh β∂+β(∂−α −ν) + ∂−β(∂+α −µ)= 0 ,cosh2 β ∂+∂−β −tanh β (∂−α −ν)(∂+α −µ) = 0 ,(A.14)that Λ is indeed chiral, satisfying∂−Λ = 0 . (A.15)Moreover one also checks the following Poisson brackets{Λ(x), α(y)} = −(∂+α −µ) δ(x1 −y1) ,{Λ(x), β(y)} = −(∂+β) δ(x1 −y1),(A.16)which reproduce the transformations (A.10).

Thus the density Λ features all what isexpected from the Noether charge density associated with the conformal symmetry.Finally, we present here for completeness the general solution of the equations ofmotion (A.14). Along the lines of Section 2.3, it can be obtained as follows:α = (ηL + ηR) + tan−1hsinh(θL −θR)sinh(θL + θR) tan(λL −ρR)i+ µx+ + νx−,cosh(2β) = cosh(2θL) cosh(2θR) + sinh(2θL) sinh(2θR) cos(2(λL −ρR)),(A.17)where {ηL, λL, θL} and {ηR, ρR, θR} are arbitrary functions of x+ and x−only, respec-tively, and the three functions of each chirality are related by the equations,∂+ηL + ∂+λL cosh(2θL) = 0 ,∂−ηR + ∂−ρR cosh(2θR) = 0 .

(A.18)76

Appendix B: H-compatible sl(2) and the non-degeneracy conditionOur purpose in this technical appendix is to analyse the notion of the H-compatiblesl(2) subalgebra, which has been introduced in Section 3.3. We recall that the sl(2)subalgebra S = {M−, M0, M+} of the simple Lie algebra G is called H-compatible if His an integral grading operator, [H , M±] = ±M±, and M± satisfy the non-degeneracyconditionsKer(adM±) ∩GH∓= {0}.

(B.1)Note that the second property in this deﬁnition is equivalent to the fact that S commuteswith (H −M0). We prove here the results stated in Section 3.3, and also establish analternative form of the non-degeneracy condition, which will be used in Appendix C.Let us ﬁrst consider an arbitrary (not necessarily integral) grading operator H ofG and some non-zero element M−from GH−1.

We wish to show that to each such pair(H, M−) there exists an sl(2) subalgebra S = {M−, M0, M+} for which M+ ∈GH+1. om-mutes To exhibit the S-triple in question, we need the Jacobson-Morozov theorem, whichhas already been mentioned in Section 3.3.

In addition, we shall also use the followinglemma, which can be found in [33] (Lemma 7 on page 98, attributed to Morozov).Lemma: Let L be a ﬁnite-dimensional Lie algebra over a ﬁeld of characteristic 0 andsuppose L contains elements h and e such that [h , e] = −e and h ∈[L , e]. Then thereexists an element f ∈L such that[h , f] = fand[f , e] = 2h .

(B.2)Turning to the proof, we ﬁrst use the Jacobson-Morozov theorem to ﬁnd generators(m−, m0, m+) in G completing m−≡M−to an sl(2) subalgebra. We then decomposethe elements m0 and m+ into their components of deﬁnite grade, i.e., we writem0 =Xnmn0andm+ =Xnmn+ ,(B.3)where n runs over the spectrum of the grading operator H. Since M−is of grade −1, itfollows from the sl(2) commutation relations that[m00 , M−] = −M−and[m1+ , M−] = 2m00 ,(B.4)77

and these relations tell us that h = m00 and e = M−satisfy the conditions of the abovelemma. Thus there exists an element f satisfying (B.2), which we can write as f = Pn f nby using the H-grading again.

The proof is ﬁnished by verifying that M+ ≡f 1 andM0 ≡m00 together with M−span the required sl(2) subalgebra of G.From now on, let H be an integral grading operator. For an element M± of grade ±1,respectively, the pair (H, M±) is called non-degenerate if it satisﬁes the correspondingcondition in (B.1).We claim that if S = {M−, M0, M+} is an sl(2) for which the generators M± are fromGH±1, then the non-degeracy of the pairs (H, M−) and (H, M+) are equivalent statements.This will follow immediately from the sl(2) structure if we prove that the non-degeneracyof the pair (H, M±) is equivalent to the following equality:dim Ker(adM±) = dim GH0 .

(B.5)It is enough to prove this latter statement for a pair (H, M−), since then for a pair(H, M+) it can be obtained by changing H to −H. To prove this let us ﬁrst rearrangethe identitydim G = dim Ker(adM−) + dim [M−, G](B.6)by using the grading asdim Ker(adM−) −dim GH0 =dim GH+ −dim [M−, GH+ ]+dim GH−−dim [M−, GH0 + GH−] .

(B.7)Since both terms on the right hand side of this equation are non-negative, we see thatdim Ker(adM−) ≥dim GH0 ,(B.8)and equality is achieved here if and only ifdim GH+ = dim [M−, GH+ ]and[M−, GH0 + GH−] = GH−. (B.9)On the other hand, we can show that the two equalities in (B.9) are actually equivalent toeach other.

To see this, let us assume that the second equality in (B.9) is not true. Thisis clearly equivalent to the existence of some non-zero u ∈GH+ such that ⟨u , [M−, GH0 +GH−]⟩= {0}.

By the invariance and the non-degeneracy of the Cartan-Killing form, this isin turn equivalent to [M−, u] = 0, which means that the ﬁrst equality in (B.9) is not true.By noticing that the ﬁrst equality in (B.9) is just the non-degeneracy condition for the78

pair (H, M−), we can conclude that the non-degeneracy condition is indeed equivalentto the equality in (B.5).We wish to mention a consequence of the results proven in the above. To this letus consider a non-degenerate pair (H, M−).

By our more general result, we know thatthere exists such an sl(2) subalgebra S = {M−, M0, M+} for which M+ is from GH+1. Thepoint to mention is that this S is an H-compatible sl(2) subalgebra, as has already beensated in Section 3.3.

In fact, it is now easy to see that this follows from the equivalence of(B.1) with (B.5) by taking into account that the kernels of adM± are of equal dimensionby the sl(2) structure.79

Appendix C: H-compatible sl(2) embeddings and halvingsIn Section 3.3, we showed that, given a triple (Γ, M, H) satisfying the conditions forﬁrst-classness, conformal invariance and polynomiality (eqs. (2.6), (2.13) and (3.2-4)),the corresponding W-algebra is isomorphic to WGS , provided that H is an integral gradingoperator.

Here S = {M−, M0, M+} is some sl(2) subalgebra containing M−= M. Anatural question is what sl(2) subalgebras arise in this way, or equivalently, given anarbitrary sl(2) subalgebra, can the resulting WGS -algebra be obtained as the W-algebracorresponding to the triple (Γ, M, H), for some integral grading operator H ? Whetherthis occurs or not depends only on how the sl(2) is embedded, and it is therefore a puregroup-theoretic question.

According to Section 3.3, the sl(2) subalgebras having thisproperty are the H-compatible ones. This appendix is devoted to establishing when agiven sl(2) embedding is H-compatible, and if so, what the corresponding H is.The question of an sl(2) being H-compatible is very much related to another one,which was mentioned at the end of Section 4.2.

We noted that in some instances, ageneralized Toda theory associated to an sl(2) embedding could as well be regarded as aToda theory associated to an integral grading operator H. This means that the eﬀectiveaction of the theory is a special case of both (4.12) and (4.3) at the same time. We haveseen that this is the case when the corresponding halving is H-compatible, i.e., when theLie algebra decomposition G = (G≥1+P 12 )+(Q 12 +G0+Q−12 )+(P−12 +G≤−1) (subscriptsare M0-grades) can be nicely recasted into G = GH≥1 + GH0 + GH≤−1.

Our second problem,addressed at the end of the appendix, is to ﬁnd the list of those sl(2) subalgebras whichallow for an H-compatible halving. Clearly, an sl(2) subalgebra which possesses an H-compatible halving is also H-compatible in the above sense, but it will turn out that theconverse is not true.Let S = {M−, M0, M+} be an sl(2) subalgebra embedded in a maximally non-compact real simple Lie algebra G. For the classical algebras Al, Bl, Cl and Dl, thesereal forms are respectively sl(l+1, R), so(l, l+1, R), sp(2l, R) and so(l, l, R).

(We do notconsider the exceptional Lie algebras.) For S to be an H-compatible sl(2), one shouldﬁnd an H in G with the following properties:1. adH is diagonalizable with eigenvalues being integers,2.

H −M0 must commute with the S-triple,80

3. dim Ker(adH) = dim Ker(adM±).We remark that here the equivalence of relations (B.1) and (B.5), proven in the previousappendix, has been taken into account. Under conditions 1-3, the decompositionΓ⊥= [M−, Γ] + Ker(adM+)(C.1)holds, where Γ = GH≥1 in the (Γ, M−, H) setting, or Γ = P 12 + GM0≥1 in the sl(2) setting,respectively.

(For clarity, note that these two gauge algebras are in general not equal. )As a consequence, Jred(x) = M−+ jred(x) with jred(x) ∈Ker(adM+) is a DS gauge inboth settings, and thus the W-algebras are the same.In order to answer the question of whether an sl(2) embedding is H-compatible,it is useful to know what these embeddings actually are.

For a classical complex Liealgebra Gc, this question has been completely answered by Malcev (and Dynkin for theexceptional complex Lie algebras) [39]. The result can be nicely stated in terms of theway the fundamental vector representation reduces into irreducible representations of thesl(2):Al : the sl(2) reduction of the (l+1)-dimensional representation can be arbitrary,Bl : the (2l + 1)-dimensional representation of Bl reduces in such a way that the multi-plicity of each sl(2) spinor appearing in the reduction is even,Cl : the 2l-dimensional representation of Cl reduces in such a way that the multiplicityof each sl(2) tensor appearing in the reduction is even,Dl : same restriction as the Bl series: the spinors come in pairs.The above conditions are necessary and suﬃcient, i.e., every possible sl(2) content sat-isfying the above requirements actually occurs for some sl(2) embedding.

Moreover, forthe classical complex Lie algebras, the way the fundamental reduces completely speciﬁesthe sl(2) subalgebra, up to automorphisms of the embedding Gc [39].The above description of the sl(2) embeddings remains valid for the maximally non-compact classical real Lie algebras, except the last statement. First of all, this means thatthe above restrictions apply to the possible decompositions of the fundamental under thesl(2) subalgebras in the real case as well.

It is also obvious that those sl(2) embeddingsfor which the content of the fundemantal is diﬀerent are inequivalent.The converse81

however ceases to be true in the real case in general: inequivalent sl(2) subalgebras canhave the same multiplet content in the fundamental of G. The answer to the problem ofH-compatibility will in fact be provided by looking more closely at the decomposition ofthe fundamental of G under the sl(2) subalgebra in question, as will be clear below.As an immediate consequence of condition 2, H −M0 is an sl(2) invariant and canonly depend on the value of the Casimir. If, in the reduction of the fundamental of G, aspin j representation occurs with multiplicity mj, the sl(2) generators ⃗M and H can bewritten⃗M =Xj⃗M (j) × Imj,(C.2a)H = M0 +XjI2j+1 × D(j),(C.2b)where In denotes the unit n × n matrix, and the D(j)’s are mj × mj diagonal matrices.Hence, within each irreducible representation of sl(2), H is equal to M0 shifted by aconstant.

Obviously, this is also true in the adjoint representation and, in turn, thisimplies that adH takes the value zero at most once in each sl(2) multiplet in the adjointof G.From condition 3, adH must take the value zero exactly once, i.e., each sl(2)representation must intersect Ker(adH) exactly once. In particular, the sl(2) singletsmust be adH-eigenvectors with zero eigenvalue.The trivial solution H = M0 exists whenever adM0 is diagonalizable on the integers,i.e., when the reduction of the fundamental of G is either purely tensorial or purelyspinorial.From now on, we suppose that the reduction involves both kinds of sl(2)representations.1) Al algebras.The problem for the Al series is simple to solve since, in this case, an H always exists.As a proof, we explicitly give an H which fulﬁlls all the requirements.

In (C.2b), we setD(j) = λ · Imjif j ∈N,(λ + 12) · Imjif j ∈N + 12,(C.3)where λ is a constant that makes H traceless. In order to show that the H so deﬁnedhas the required properties, we recall that for the Al algebras, the adjoint representation82

is obtained by tensoring the fundamental with its contragredient. As a result, the rootsare the diﬀerences of the weights of the fundamental (up to a singlet) and we haveadH = adM0 + [D(j1) −D(j2)],(C.4)where j1 and j2 are the spins of the states in the fundamental representation from whicha given state in the adjoint representation is formed.

That the conditions 1-3 are satisﬁedis obvious from the fact that adH = adM0 on tensors and adH = adM0 ± 12 on spinors,with + 12 occurring as many times as −12.It should be pointed out that (C.3) is by no means the only solution. Since in theproduct j1 × j2, the highest weights have an M0-eigenvalue at least equal to |j1 −j2|,another solution is given by D(j) = (λ + j) · Imj.2) Cl algebras.For the symplectic algebras, the adjoint representation is obtained from the symmetricproduct of the fundamental with itself and we therefore haveadH = adM0 + [D(j1) + D(j2)].

(C.5)Since the symmetric product of a tensor with itself produces a singlet, which must belongto Ker(adH), we have 2D(t) = 0 for every integer j = t. Hence in the fundamentalrepresentation, H = M0 on tensors. Similarly, the symmetric product of a spinor withitself always produces a triplet, one member of which must belong to Ker(adH).

Thisimplies that the diagonal entries of 2D(s) are either 0 or ±1, for every half-integer j = s.However D(s) cannot have a zero on the diagonal, because adH would not be integral onthe representations contained in s × t. Therefore, in the fundamental, H = M0 ± 12 onspinors.Let us now look at the ms spinor representations of spin s, say s1, s2, . .

., sms. Theproduct si×sj of any two of those contains a singlet, and that implies D(si)+D(sj) = 0.This equality must hold for any pair of spin s representations, which is impossible unlessms ≤2.Let us consider the restriction gs of the symplectic form to the spin s representations.The restricted form is non-degenerate, because the original non-degenerate metric isblock-diagonal with respect to the eigenvalues of the sl(2) Casimir.83

If ms = 1, then the H given by M0± 12 ·I on the unique spin s representation, shouldbe in the symplectic algebra: gsH +Htgs = 0. Since M0 is already symplectic, we requirethat the identity be symplectic, which is impossible for a non-degenerate form.

Hencems must be 2.If ms = 2, H −M0 and gs look like (in the basis where M0 and H are diagonal)H −M0 = ± 1200−12,gs =ab−btc,(C.6)where the blocks a and c are antisymmetric. H −M0 being symplectic leads to a = c = 0.To summarize, for an integral H to exist, the sl(2) embedding must be such that: (i)the multiplicity of any spinor representation in the fundamental of G is 2, (ii) if (s, s′) issuch a pair of spinors, they must be the dual of each other with respect to the symplecticform.

If these two conditions are met, then H is given in the fundamental byH = M0on tensors,M0+/−12on a pair of spinors s/s′. (C.7)Conditions 1-3 are satisﬁed since (C.7) implies adH = adM0 on singlets, adH = adM0 ±(1or 0) on tensors and adH = adM0 ± 12 on spinors.3) Bl and Dl algebras.The analysis here is similar to what has been done in 2), and we can therefore go throughthe proof quickly.For the orthogonal algebras, the adjoint is got from the antisymmetric product ofthe fundamental with itself and we still haveadH = adM0 + [D(j1) + D(j2)].

(C.8)The antisymmetric product of a tensor (spinor) with itself produces a triplet (singlet),so that with respect to the symplectic algebras, the situation is reversed in the sensethat the tensors and the spinors have their roles interchanged: H = M0 ± 12 on tensors,H = M0 on spinors and mt ≤2 for any tensor representation of spin t.If as in 2), we look at the restriction gt of the orthogonal metric to the spin ttensors, we have mt = 2 on account of the non-degeneracy of gt. From this, we get at84

once that there can be no solution for the Bl algebras. Indeed, the fundamental beingodd-dimensional, at least one tensor representation must come on its own.On the 2(2t + 1)-dimensional subspace made up by the two spin t tensors, H −M0and gt take the formH −M0 = ± 1200−12,gs =abbtc,(C.9)where a and c are now symmetric.

Requiring that H −M0 be orthogonal, we againobtain a = c = 0.Therefore, for the orthogonal algebras, we get the following conclusions. There isno solution for the Bl series if the sl(2) embedding is not integral.

As to the Dl series,the sl(2) embedding must be such that: (i) every tensor in the fundamental of G has amultiplicity equal to 2, (ii) if (t, t′) is such a pair of tensors, they must be the dual of eachother with respect to the orthogonal metric. In this case, H is given in the fundamentalbyH =M0+/−12on a pair of tensors t/t′,M0on spinors.

(C.10)Summarizing the analysis, the H-compatible sl(2) embeddings are the followingones:Al : any sl(2) subalgebra,Bl : only the integral sl(2)’s,Cl : those for which each spinor occurs in the fundamental of Cl with a multiplicity0 or 2, the pairs of spinors being symplectically dual,Dl : those for which each tensor occurs in the fundamental of Dl with a multiplicity0 or 2, the pairs of tensors being orthogonally dual.The reader may wish to check that the above results are consistent with the isomorphismsB2 ∼C2 and A3 ∼D3.We now come to the second question alluded to at the beginning of this appendix,85

namely the problem of H-compatible halvings. From the deﬁnition, an sl(2) subalgebraallows for an H-compatible halving if in addition to conditions 1-3 one also has4.

P 12 + G≥1 = GH≥1, and P−12 + G≤−1 = GH≤−1.In particular, this fourth condition implies GM00⊂GH0 . So we readily obtain that H andM0 must satisfyadH = adM0,ontensors,(C.11)since we know, from the previous analysis, that adH −adM0 is a constant in everyrepresentation (condition 2).

Therefore, we can simply look at those solutions of the ﬁrstproblem which satisfy (C.11) and check if condition 4 is fully satisﬁed or not. We getthat the sl(2) embeddings allowing for an H-compatible halving are as follows:Al : any sl(2) subalgebra.

There are only two solutions for H given by setting in(C.2b): D(j) = (λ ± ǫ(j)) · Imj with ǫ(j) = 0/ 12 for a tensor/spinor,Bl : only the integral sl(2)’s with H = M0,Cl : only the integral sl(2)’s,Dl : the integral sl(2)’s, and those for which the fundamental of Dl reduces intospinors and two singlets, with H given by (C.10).86

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